Logic As the Universal Science. Bertrand Russell's Early

Anssi Korhonen

Logic as the Universal Science: Bertrand Russell’s Early Conception of Logic and Its Philosophical Context

Philosophical Studies from the University of Helsinki 18

Filosofisia tutkimuksia Helsingin yliopistosta Filosofiska studier från Helsingfors universitet Philosophical Studies from the University of Helsinki

Publishers: Department of Philosophy Department of Social and Moral Philosophy P.O. Box 9 (Siltavuorenpenger 20 A) 00014 University of Helsinki Finland

Editors: Marjaana Kopperi Panu Raatikainen Petri Ylikoski Bernt Österman Anssi Korhonen

Logic as the Universal Science: Bertrand Russell’s Early Conception of Logic and Its Philosophical Context ISBN 978-952-10-4406-9 (paperback) ISBN 978-952-10-4407-6 (pdf, http://ethesis.helsinki.fi) ISSN 1458-8331 Vantaa 2007 Dark Oy Acknowledgements

I owe very special thanks to Professor Gabriel Sandu for his continuous and very concrete support of this project, which has been rather slow in unfolding. I wish to thank my supervisor, Professor Leila Haaparanta, who introduced me to the serious study of the history of modern logic and analytic philosophy. Professor Nicholas Griffin, Professor André Maury and Dr. Patrick Sibelius provided invaluable comments on earlier drafts of the manuscript. I owe special thanks to the wonderful, and wonderfully heterogeneous, collective that is constituted by the members of the Russell-l discussion forum.

I also wish to extend my thanks to Dr. Panu Raatikainen, Mr. Simo Rinkinen and Mr. Max Weiss for discussions, comments and concrete advice. I owe very special thanks to my dear friends and colleagues, Dr. Markku Keinänen and Mr. Pekka Mäkelä, for co-operation, support and innumerable other things. I am indebted to my colleagues and room- mates, Dr. Pauliina Remes and Mr. Fredrik Westerlund, for providing a perfect atmosphere in which to fight the last battle against the recalcitrant manuscript. The Head of our Department, Dr. Thomas Wallgren, provided very concrete help during the final stage of this study. In preparing the manuscript for print, I have received excellent editorial help from Mrs. Auli Kaipainen.

Last, but in many ways first, I wish to thank Niina for her unfailing support, encouragement, sympathy and patience.

The financial support provided for this work by the Finnish Cultural Foundation, Emil Aaltosen Säätiö, The Finnish Academy and University of Helsinki is gratefully acknowledged.

I dedicate this work to the memory of my parents.

Helsinki, November 2007 Anssi Korhonen

Acknowledgements 5 Contents 7

Introduction 15

1 Preliminary Remarks on Russell’s Early Logicism 27

1.0 Introduction 27 1.1 Different Logicisms 30 1.2 Analyticity and Syntheticity 32 1.2.1 Preliminary Remarks 32 1.2.2 Kantian Analyticity 33 1.2.3 Logical Empiricism and Analyticity 35 1.2.4 Analyticity in Frege and Russell 36 1.3 The Pursuit of Rigour 42 1.3.1 The Mathematical Context 42 1.3.2 Why Rigour? 45 1.3.3 Epistemic Logicism 48 1.3.4 What is Really Involved in Rigour 59 1.4 Conclusions: Russell and Kant 63

2 Kant on Formal-logical and Mathematical Cognition 71

2.0 Introduction 71 2.1 Kant’s Programme for the Philosophy of Mathematics 73 2.1.1 Preliminary Remarks 73 2.1.2 The Leibnizian Background 74 2.1.3 Kant on Analytic and Synthetic Judgments 78 2.1.4 The Reasons behind Kant’s Innovations 80 2.1.4.1 Concepts and Constructions 80 2.1.4.2 The Containment Model for Concepts 83 2.1.4.3 A Comparison with Frege 86 8 Contents

2.1.4.4 Kant on Philosophical and Mathematical Method 88 2.1.4.5 Summary 90 2.2 Constructibility and Transcendental Aesthetic 92 2.3 The Semantics of Geometry Explained 94 2.3.1 Constructions in Geometry 94 2.3.2 Geometry and Space 101 2.4 Constructions in Arithmetic 105 2.5 The Applicability of Mathematics 108 2.6 Pure and Applied Mathematics 109 2.7 The Apriority of Mathematics According to Kant 112 2.8 Conclusions 117

3 Russell on Kant 121

3.0 Introduction 121 3.1 Russell on the Nature of the Mathematical Method 124 3.1.1 “The Most Important Year in My Intellectual Life” 124 3.1.2 Russell and Leibniz 125 3.1.3 Russell and Peano 127 3.1.4 The Concept of Deductive Rigour 132 3.1.4.1 General Remarks 132 3.1.4.2 Pasch on Rigorous Reasoning 135 3.1.4.3 The Logicization of MathematicalProof 143 3.1.5 Russell on Rigorous Reasoning 145 3.1.5.1 Self-Evidence and Rigour 145 3.1.5.2 Different Sources of Self-Evidence 147 3.1.5.3 Logical Self-Evidence 150 3.1.5.4 Poincaré on Intuition and Self-Evidence 149 Contents 9

3.2 Kant and Misplaced Rigorization 156 3.2.1 Russell and Kant on Mathematical Reasoning 156 3.2.2 Some Remarks on the Standard View 160 3.3 Russell’s Criticisms of Kant 166 3.3.1 Russell on Intuitions 166 3.3.2 Russell’s Kantian Background 169 3.3.3 Quantity in the Principles 173 3.3.4 Propositional Functions in the Principles 177 3.3.5 Against Russell: the Notion of Intuition Again 180 3.4 Summary 186 3.5. The Role of Logicism 189 3.5.1 Hylton on the Role of Logicism 189 3.5.2 Criticism of Hylton’s Reconstruction 193 3.6 Russell’s Case against Kant 203 3.6.1 The Standard Picture of TranscendentalIdealism 203 3.6.2 The Implications of the Standard Picture 207 3.6.2.1 Kant’s “Subjectivism” 207 3.6.2.2 Moore against Kant 208 3.6.3 The Relativized Model of the Apriori 212 3.6.3.1 Preliminary Remarks 212 3.6.3.2 Three Direct Arguments Againstthe R-Model 213 3.6.3.3 Three Indirect Arguments againstthe R-Model 222 3.6.3.3.1 The Consequences of the R-Model 222 3.6.3.3.2The Argument from Necessity 224 3.6.3.3.3 Another Argument from Necessity 258 3.6.3.3.4The Argument from Truthand the Argumentfrom Universality 259 10 Contents

4 Logic as the Universal Science I: the van Heijenoort Interpretation and Russell’s Conception of Logic 265

4.0 Introduction 265 4.1 “Logic as Calculus and Logic as Language” 266 4.2 van Heijenoort’s Distinction 271 4.2.1 The Technical Core of the Model-Theoretic Conception 271 4.2.2Two Conceptions of Generality 276 4.2.3 The Technical Core of the Universalist Conception 280 4.3 The Philosophical Implications of the Universalist Conception 283 4.3.1 “No Metaperspective” 283 4.3.2 Two Senses of “Interpretation” 287 4.3.3A Flowchart 292 4.4 Russell’s Notion of Proposition 299 4.4.1 Preliminary Remarks 299 4.4.2 Moore’s Theory of Judgment 302 4.4.3 Predication 309 4.4.3.1 Moore on Predication 309 4.4.3.2 Russell’s Criticisms of Moore 311 4.4.4 Moorean and Peanist Elements in Russell’sTheory of Propositions 314 4.4.5 The Notion of Term 315 4.4.6 The Problem of Unity 318 4.4.6.1 A Fregean Perspective on Predication 318 4.4.6.2 Comparing Frege and Russellon Predication 321 4.4.6.3 Problems with Russell’s Account of the Problem of Unity 325 4.4.6.4 Propositions as Facts 327 4.4.6.5 A Way Out for Russell? 330 4.4.7 Russell’s Notion of Assertion 333 4.4.8 Peano’s Logic 338 Contents 11

4.4.8.1 Why Peano is Superior to Moore 338 4.4.8.2 Predication and Propositional Functions 341 4.4.9 The Theory of Denoting 345 4.4.9.1 Introducing Denoting Concepts 345 4.4.9.2 Why Denoting Concepts are Needed 348 4.4.9.3 Denoting Concepts and PropositionalFunctions 352 4.4.10 Russell’s Analysis of Generality 355 4.4.10.1 General Remarks 355 4.4.10.2 Formal Implications 357 4.4.10.3 The Propositions of Logic Again 359 4.4.11 Comparing Russell and Frege on the Constitution of Propositions 363 4.4.12 Russell’s Account of Variables 369 4.4.13 Conclusions 374 4.5 Russell’s Version of the Universalist Conception of Logic 375 4.5.1 Preliminary Remarks 375 4.5.2 Russell’s Alleged Anti-Semanticism 380 4.5.2.1 The Fixed Content Argumentand the Argumentfor Uniqueness 380 4.5.2.2 What is wrong with the UniquenessArgument 384 4.5.2.2.1 Did Russell Have a “Calculusof Logic”? 387 4.5.2.2.2 Further Remarks on Russell’s Conception of Calculus 391 4.5.2.2.3Reasoning about Reasoning 392 4.5.2.2.4 The Justification of Logic 395 4.5.2.3 What is wrong with the FixedContent Argument 400 4.5.2.3.1Justification and 12 Contents

SemanticExplanation 400 4.5.2.3.2Frege on the Semantic Justificationof Logic 403 4.5.2.3.3Calculus for Logic and the Science of Logic 405 4.5.2.3.4Russellian Metatheory? 407 4.5.3 The True Source of Russell’s Anti-Semanticism 414 4.5.4Russell on Alternative Interpretations 417 4.5.5 “Interpretation” and Semantics 427 4.5.6 Generality and Quantification 433 4.5.6.1 Unrestricted Generality 433 4.5.6.2 Hylton on Russell on Generality 435 4.5.6.3 Criticism of Hylton’s Reading 437 4.5.7 Russell’s Concept of Truth 441 4.5.7.1 Preliminary Remarks 441 4.5.7.2 An Analogy with Frege 442 4.5.7.3 Frege’s Version of the Argumentagainst Truthdefinitions 448 4.5.7.4 The Metaphysics of Truth 450 4.5.7.5 Truth-Primitivism and Truth-Attributions 453 4.5.7.6 Use of the Truth-Predicate 455 4.6 Russell’s Conception of Mathematical Theories 459 4.6.1 General Remarks 459 4.6.2 The Frege-Hilbert Controversy 460 4.6.3The Russell-Poincaré Controversy 465 4.6.4 Philosophical and Mathematical Definitions 470 4.6.5 What did Russell Really say about MathematicalTheories 472 4.6.6Russell and Abstract Axiomatics 475 4.7 Conclusion 479 Contents 13

5 Logic as the Universal Science II: Logic as a Synthetic Apriori Science 481

5.0 Introduction 481 5.1 Hylton on Russell’s Commitment to the Universalist Conception 483 5.2 Kant on Formal Logic 496 5.2.1 Preliminary Remarks 496 5.2.2 Formality and the Analyticity-Constraint on Kant 499 5.2.3 Analyticity and Apriority 507 5.3 The Bolzanian Account of Logic 509 5.3.1 Preliminary Remarks 509 5.3.2 The Basic Assumptions behind the BolzanianAccount 511 5.3.3 Russell’s Version of the Bolzanian Account 517 5.3.3.1 Russell on Form and Content 517 5.3.3.2 Russell’s Version of the Schematic Account of Logical Form 526 5.4 The Universalist Conception and Logical Constants 534 5.5 Universality and the Normative Conception of Logicality 539 5.6 Russell and the Normative Conception of Logicality 551 5.7 Russell and the Descriptive Account of Logicality 557 5.7.1 The Development of Russell’s Views on Generality 557 5.7.2 Russell’s Account of Logical Generality 563 5.8 Logic as Synthetic 567 5.9 The Demarcation of Logic 572 5.9.1 The Propositions of Logic as Formal Implications 572 5.9.2 Russell on Valid Inference 574 5.9.3Russell on Lewis Carroll’s Puzzle 579 5.9.4 The Bolzanian Account of 14 Contents

Generality and Russell’s Ontology 583 5.10 The Apriority of Logic 587 5.11 Concluding Remarks 591

Bibliography 593 Introduction

The present work concerns Bertrand Russell (1872-1970) and the conception of logic that underlies the early version of his logicist philosophy of mathematics. The views examined here are those that are found in the Principles of Mathematics, work published in 1903. The single most important event leading to this work was Russell’s participation in the International Congress of Philosophy in Paris in the early August of 1900. The Paris Congress was of paramount importance to Russell’s philosophical development – he was later to describe it as the most important event in the most important year of his intellectual life – because it was there that he met Giuseppe Peano (1858-1932), an Italian mathematician, and first learned about mathematical logic in any serious sense of that term. Principles in turn is an exposition of Russell’s initial vision of logicism, which he formulated as a result of reflecting on the arithmetization of analysis and geometry – “modern mathematics” as he used to call it – and the methodological consequences of the new logic. There are three reasons why I have singled out this period in Russell’s philosophy for a detailed examination. Firstly, there is the simple point that the views propounded in the Principles and related works are of considerable intrinsic interest and deserve to be examined in their own right.1 Secondly, the early version of logic and logicism constitutes the essential background for many of the later theories for which Russell is best remembered. Typically, these came into existence only after 1903: the theory of types (although a rudimentary formulation of the simple theory is already worked in an Appendix to the Principles), the theory of definite descriptions and the notion of an incomplete symbol that goes together with it, the

1 Not that I am the first to discover this. The early realist (or post- idealist) Russell has been the subject of much first-rate scholarship in the recent years: see, for example, Hylton (1990a), Hager (1994) Landini (1998), Linsky (1999), Makin (2000), Stevens (2005) and such collections of essays as Irvine and Wedeking (1993), Monk and Palmer (1996) and Griffin (2003). 16 Introduction multiple relation theory of judgment and Russell’s version of the correspondence theory of truth, to mention the most important cases. These, though, are not theories that simply came after the Principles; each of them is an eventual response to some difficulty or difficulties inherent in the initial vision of logic and logicism. Thirdly, examination of Russell’s views contributes to our understanding of early analytic philosophy and the development of modern logic. Russell did not invent either of these; mathematical logic was originated by Frege in 1879 – or, possibly, by Boole in 1847 or by Bolzano in 1837 – and analytical philosophy may owe its origin to Wittgenstein, who effected the linguistic turn in the Tractatus – or, possibly, the young Moore who switched from idealism to analysis in the late 1890s, or by Frege who at least anticipated the linguistic turn in 1884 with his context principle, or by Bolzano who anticipated in the 1830 many of the things that were worked out only much later by the likes of Frege, Carnap and Tarski. Be these questions as they may, Russell was at least responsible, more than anyone else, for creating the unique combination of mathematical logic and philosophical thought that characterizes what may be called “analytical modernism”.2 The philosophical context of his early logicism and the universalist conception of logic provide an excellent illustration of what that combination may amount to in practice. As for modern logic and its philosophical underpinnings, it is a claim often endorsed that Russell’s views represent the so-called universalist conception of logic. There is, though, no consensus on how this view ought to be understood or, indeed, whether some such view is legitimately attributed to Russell. In the present work these interpretative issues will be examined in their historical and philosophical context. Whatever else may be said about my conclusions, I hope to have made it probable that there is no simple formula for interpreting Russell’s views on the nature of formal logic.

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2 I borrow this term from Skorupski (1993). Introduction 17

Russell had been working on the philosophy of mathematics from the very beginning of his philosophical career – he went up to Cambridge in 1890, completed Part I of Mathematical Tripos in 1893, Part II of the Moral Sciences Tripos the following year and won a six- year Prize Fellowship at Trinity College with an essay on the foundations of geometry in 1895 – and his first professional efforts in this field are firmly in the idealist tradition. The chief influences on his philosophy at that time were F. H. Bradley (1846-1924), a somewhat enigmatic figure who had just “done as much in metaphysics as is humanly possible” with the publication of Appearance and Reality in 1893 (the characterization is G. F. Stout’s), James Ward (1843-1925), one of Russell’s teachers at Cambridge, and J. M. E. McTaggart (1866-1925), Russell’s early philosophical inspiration and a fellow Apostle at Cambridge3. But the plan for future work that Russell worked out after graduation was an original one, though the outlines were what one might expect from a neo-Hegelian philosopher. The underlying idea was to work the way towards the Absolute, and this was to be accomplished “dialectically”; beginning with the results established in special sciences, progress would take place through the resolution of contradictions that inevitably turn up when one examines these results. The starting- point for the dialectical process was to be found in mathematics, apparently because, being abstract, it was the science that was as far away from the Absolute as one could get while still retaining vestiges of truth. For anything short of the Absolute, if treated as independent and self-contained, is an abstraction and involves a measure of untruth (“contradiction”, to use the idealist locution). Working towards the absolute thus means: to modify the one-sided expression of truth found in some special science by adding new layers to it, or by embedding it in something more inclusive; the movement would thus be from a system that is exceedingly abstract (arithmetic)

3 I owe this characterization of Russell’s and McTaggart’s relationship to Nicholas Griffin. 18 Introduction through a series of dialectic steps (geometry, physics, psychology) towards the metaphysics that would reveal the Absolute for what it really is, the ultimate system that is maximally concrete in that it includes everything within it and also maximally real, because in it all contradictions would vanish to give way to one harmonious, self- consistent system. This may look like a rather inauspicious start for a serious philosophy of mathematics. And Russell did find out that he could make very little progress, dialectical or otherwise, as long as he remained an idealist; which meant, as long as he retained the principles underpinning the dialectical method. Having been an idealist, he turned to a rather extreme form of realism, describable, perhaps, as a version of platonism. This development, in which he was aided by his colleague, G. E. Moore (1873-1958), did not change his research interests, for he continued to work on the “principles” of mathematics. And yet, though he was to argue that acceptance of Moore’s metaphysics “brought an immediate liberation from a large number of difficulties”, he did not really achieve much until he learned to appreciate what such mathematicians as Cantor, Dedeking and Weierstrass had achieved on the Continent and how this could be illuminated by drawing on the resources provided by mathematical logic. That mathematical reasoning is really a matter of formal logic may not strike us as a particularly extraordinary view. Yet, what he learned from Peano was a revelation to Russell, in particular since Peano’s symbolic logic showed what that claim about mathematical reasoning could mean in practice. Russell’s logicism, however, was not just the view that formal logic is relevant to understanding mathematical reasoning; he believed he could demonstrate that what he called “pure mathematics” is reducible to logic. What, exactly, this means is something that he never succeeded in explaining very clearly – part of what is involved in the Russellian logicism is clear enough, part of it much less so – but a working characterisation might be along the following terms. Pure mathematics is the totality of mathematical Introduction 19 theories formulated as sets of axioms or basic propositions from which theorems are derived, and “logicism” refers to the thesis that all the concepts, propositions and reasonings that are needed in the reconstruction of pure mathematics are logical in character. There is in the present essay very little about the technical issues surrounding logicism that inevitably constitute much of its content. Instead, my focus will be on the philosophical context of Russell’s logicism, with a view of providing a detailed account of the conception of logic that informs it. I shall begin this task by considering, in Chapter 1, certain general issues relating to Russell’s logicism. Firstly, I will show how the familiar distinction between analyticity and syntheticity can be used to throw light upon Russell’s logicism. Secondly, I will argue, that his version of logicism should be seen as an attempt to improve our understanding of the content of mathematics, of what is really involved in mathematical concepts, propositions, reasonings and, eventually, in mathematical theories, rather than as a contribution to the epistemology of mathematics, which is still a fairly common assumption. Logicism is thus a typical philosophical thesis in that it makes a number of claims about the true nature of something, in this case about the nature of mathematical concepts (they are in fact concepts belonging to logic), truths (they are logical truths) and reasonings (there are no irreducibly mathematical modes of reasoning, since all valid deduction is a matter of logic). Like typical philosophical theses, Russell’s views on the nature of mathematics have a critical dimension to them. And, like most variants of logicism, their target is Kant’s conception of mathematics, which was still very much a live option at the time when Russell entered philosophy, something that can be readily seen, for instance, by considering his plans for the dialectic of the sciences. Kant’s remarks on mathematics serve a largely critical purpose, that of demonstrating that there is an unbridgeable methodological gap between mathematics and philosophy. A positive programme for a philosophy of mathematics can nevertheless be gleaned from them. Kant’s programme, as it may be called, consists of two parts. Firstly, he 20 Introduction argued for a certain representation-theoretic view, namely that the content and use of mathematical representations is not grounded in mere concepts but is based on their construction in pure intuition; whatever else this may imply, it means at least that mathematical judgments are synthetic rather than analytic. The second part of Kant’s programme is epistemic; it consists in an extended argument purporting to show how mathematics, understood in accordance with the representation-theoretic views, could be known apriori. To explain how this is possible, Kant ended up defending transcendental idealism, or the doctrine that regards mathematical truth as being concerned with – or, as the critics often say, confined to – appearances, rather than things in themselves. Kant’s representation-theoretic views have a negative corollary. Some of his predecessors, most notably Leibniz, had made rather striking claims about formal logic and what it could do in the advancement of human knowledge. Kant believed none of this, arguing that by attending to the logical forms recognised in the traditional logic of terms – basically the simple subject-predicate form “A is B” – one could see that these are inadequate for the representation of any non-trivial content or genuine piece of reasoning. These representation-theoretic issues will be taken up in Chapter 2, in which Kant’s programme will be discussed from a broadly Russellian perspective. When Russell championed formal logic, he did not act as a spokesman for a movement urging a return to traditional logic; he considered the old formal logic every bit as sterile as Kant had done. What he advocated was mathematical logic, the entirely new type of logic which had begun to unfold in the latter half of the nineteenth century with the pursuit of mathematical rigour. Russell was not slow to derive anti-Kantian conclusions from the new type of logic. As he saw it, Kant’s representation-theoretic views were based on no more than mathematical and logical ignorance. That is, he argued that Kant’s views had been rendered obsolete by developments which were taking place in mathematics and logic: once a sufficiently rich Introduction 21 stock of new logical forms has been introduced, at least part of the role that Kant had given to pure intuition could be taken over by logic. And if, furthermore, Russell’s version of logicism is correct, then all of mathematical content can be reconstructed from a purely logical foundation, and Kant’s theory must be discarded. Although the second part of Kant’s programme is about explaining mathematical knowledge, the logicist Russell was not so much interested in this aspect of it as he was in the model of philosophical explanation that constitutes Kant’s transcendental idealism. As Russell reads it, Kant’s theory seeks to explain the traditional notion of apriori knowledge with the help of what I will call the relativized model of the apriori (r-model). Philosophers have traditionally argued for a number of consequences for apriority, and these are also accepted by Kant: if something is knowable apriori, it must be true, necessary and strictly universal. According to r-model, Kant’s explanatory strategy accounts for the presence of these characteristics by tracing them to certain standing features of the human mind. The fundamental flaw in this, Russell argued, is that it misconstrues the content of the apriori: it does not allow mathematics to be really true, or really necessary, or really universal. Reading what Russell has to say about Kant, one’s first impression is likely to be that it is little more than an exceedingly crude version of the well-known psychologistic reading of transcendental idealism. This impression is not as such wrong, for Russell was inclined to interpret Kant in a way that arguably ignores some of the finer details of transcendental idealism. However, once we articulate the consequences of the r-model, we shall find that the issues involved are far from simple; considering Kant in the light of the r-model will help us to raise questions about truth, necessity and universality that are genuinely non-trivial. Russell’s interpretation of this part of Kant’s programme will be discussed in chapter 3. The gist of Russell’s anti-Kantianism is the substitution of logic for pure intuition as the true source of mathematical truth. Since the propositions of mathematics are synthetic and apriori, logicism 22 Introduction implies that these features are found in logic as well. More specifically, I will argue that Russell understands syntheticity and apriority along the following lines:

x syntheticity logic has content x apriority logic complies with the constraints revealed through an examination of the r-model

It is in this way that we can use Russell’s anti-Kantianism as a clue to his thinking about the nature of logic. The question is: What must formal logic be like to have the role that Russell assigns to it in his anti-Kantian argument? This interpretative strategy was first suggested by Peter Hylton in his Russell, Idealism and the Emergence of Analytic Philosophy; at any rate, he was the first to develop it in detail. The present essay can therefore be regarded as continuing, in a specific respect, Hylton’s work on Russell. Although the underlying idea has been taken over from Hylton, my interpretation differs from his both as regards the content of Russell’s anti-Kantian argument and the consequences that this has for his understanding of logic. In articulating Russell’s conception of logic, Hylton relies on an interpretative strategy that has been extensively discussed in recent literature on the history of modern logic and early analytic philosophy. It was first proposed by Jean van Heijenoort in his classic paper, “Logic as Calculus and Logic as Language”, and has since been studied and refined by several scholars, including Burton Dreben, Jaakko Hintikka, Warren Goldfarb, and Thomas Ricketts, to mention just a few salient names. The basic underlying idea of the van Heijenoort interpretation is that we can gain important insight into the early history of modern logic – roughly, from Frege’s Begriffsschrift of 1879 until the 1930s – if we consider it in the light of two radically different ways of looking at logic. One of them, the conception of logic as calculus, builds on broadly model-theoretic conceptualisations, whereas the other view, logic as language, sees logic more like a ready-made language within which all rational discourse takes place, Introduction 23 or at least the skeleton of such a language. Hylton’s main point is that Russell’s commitment to the universalist conception of logic is to be understood in the light of his anti-Kantianism, i.e., that his criticisms of Kant have the force he took them to have only on the assumption that his understanding of logic was in line with the universalist conception, rather than the model-theoretic one. In Chapter 4 I will examine the early Russell’s views on logic in the light of the van Heijenoort interpretation. I think it is undeniable that his conception of logic is correctly describable as “universalist” in an appropriate sense of that term. I will argue, however, that many of the features that the proponents of this interpretation attribute to the universalist conception ought to be treated with considerable scepticism. This is a general conclusion about the interpretative strategy – and one that has been recently endorsed by several scholars. As for the specifics of Russell’s views, I will defend two conclusions. Firstly, to the extent that his conception of logic was universalist, this should be seen as a consequence of his metaphysical construal of the subject-matter of logic, which brings to the fore the notion of proposition. The consequences of this perspective can be seen, for instance, in his treatment of the concept of truth. Propositions in Russell’s sense, though they are truth-bearers, do not represent truth; their ontic status is closer to that of states-of-affairs. It is for this reason that he has no use for the notion of truth-condition, and to that extent lacks a semantic perspective on logic. It does not follow from this that truth has no role to play; it does, and the role is an important one, but it is metaphysical rather than semantic. Exactly what follows from this is a question that has to be investigated separately, and cannot be decided on the basis of general considerations such as are found in the van Heijenoort interpretation. Another concept that is in this way ambiguous is “interpretation”. The idea of semantic interpretation is seldom discussed by Russell (although he is not quite as negligent of it as one might think, reading the advocates of the van Heijenoort interpretation). Again, this does not mean that he had no room for a notion that is at least analogous to semantic interpretation. I will discuss this notion, and argue that it 24 Introduction in fact plays an important role in the early Russell’s conception of logic. Secondly, and this point relates to the first one, I will argue that Russell’s views on the relevant logical issues are not always what one would expect from an advocate of the universalist conception of logic in the light of the van Heijenoort interpretation. This means that a straightforward application of this interpretation to the early Russell is bound to return us less than the whole truth about the matter. Russell’s anti-Kantian argument has been criticized on the grounds that the introduction of an entirely new logic – logic to which mathematics can be reduced, if logicism is correct – in fact involves a change of meaning of the term “logic”; that, when Kant argued for the sterility of what he called “formal logic” while Russell maintained that mathematics in its entirety is reducible to “logic”, they were in fact talking about different subjects. Or, if this sounds too extreme, then at least one could argue that the reduction of a mathematical theory to “something else” would only show that something else to be mathematics, too, rather than logic. Henri Poincaré, the famous French mathematician, who defended the autonomy of mathematical reasoning against the advocates of the new logic, offers a good example of this line of thought, sometimes incorporating elements from the change of meaning-thesis into it. Whatever may be said for or against these ideas, it is clear that Russell himself did not see the matter in the same way as his critics; he did not think that he was simply extending the sense of “logic” so as to make it coextensive with whatever principles, concepts and modes of reasoning would turn up in the course of the rigorous reconstruction of some portion of mathematics. On the contrary, he clearly believed that logicism was right and the Kantians wrong about the relevant representation-theoretic issues. Evidently, this dispute can be considered philosophically exciting only if it is assumed that both parties have more or less settled views on the criteria for the demarcation of logic, i.e., for distinguishing logic from what is not logic. In chapter 5 I will discuss this aspect of Russell’s conception of logic, arguing that we can gain better Introduction 25 understanding of his universalist conception of logic – better, that is, than that provided by the van Heijenoort interpretation – if we consider how Russell’s understanding of logicality differs from Kant’s; that is, if we consider how the two differ over the content-character and apriority of logic. The present essay is of course seriously incomplete. As I indicated at the beginning of this introduction, my concern is with Russell’s initial vision of logicism. What came after is well-known, at least in outline, from the secondary literature. Russell’s Paradox, and others, showed that something was seriously wrong about the logic underpinning the claims of logicism. And, quite independently of the Paradox, Russell’s philosophy of logic was in a state of flux, for most of the logical concepts that he had worked out in the Principles presented rather formidable problem; if a reminder is needed, the notions of denoting and of denoting concepts will serve the purpose well. Russell’s initial vision of logic and logicism is characterized by certain simplicity, and this was certainly lost, when he fought his way towards Principia. Nevertheless, much of the universalist conception was retained and it certainly continued to exercise its influence upon Russell’s thought, probably as long as he thought seriously about the nature of logic.

Chapter 1 Preliminary Remarks on Russell’s Early Logicism

1.0 Introduction

“The nineteenth century, which prided itself upon the invention of steam and evolution”, Bertrand Russell wrote in an early essay, “might have derived a more legitimate title to fame from the discovery of pure mathematics.”1 Russell was not alone in thinking that the discovery of pure mathematics was essentially a matter finding out what is involved in mathematical reasoning. It is an eminently reasonable thought that deductive reasoning – the kind of reasoning that one uses, for instance, in mathematical proofs – should be a matter of logic. This, though, was by no means self-evident at the time Russell wrote the passage.2 There is a simple reason why that should have been so: logic in the forms in which it existed before, say, Frege and Peano, simply did not have any serious practical applications. The discoverers of pure mathematics could thus set traditional, Aristote- lian logic aside, because it was irrelevant to their concerns; and the new logic which they created did not bear much resemblance to the old one. To bring logic to bear on mathematics – not to say anything about such exotic views as logicism, or the view that pure mathematics is just an extension of logic – one had to create that logic first. In his early writings Frege, one of the pioneers, sometimes issued the complaint that the most refined logic of his time (apart from his own concept-script), namely Boole’s algebraic logic, could not be used for the analysis of real-life mathematical reasoning. Boolean logic, he admitted, could capture some logical forms. However, it ignored completely the problem of representing the content of mathematical propositions.3 It was for this purpose that Frege created his own

1 Russell (1901a, p. 366). 2 I do not mean to suggest that this view is self-evident now. 3 See Frege (1880/81; 1882). 28 Ch.1 Preliminary Remarks on Russell’s Early Logicism concept-script. Analysing conceptual content into function and introducing exact rules for the sort of reasoning that involves multiple generality, he could analyse and reconstruct reasonings which were far beyond the expressive power of Boole’s logic. In the same way, Russell remarked that 19th century logicians had invented “a new branch of logic, called the Logic of Relatives, [...] to deal with topics that wholly surpassed the powers of the old logic”.4 Taking into account the extent to which the birth of modern logic was a matter of invention, it can be seen that even though it is common to regard logicism as a paradigmatic reductive thesis, it might be more correct to say that it was not so much a reduction of some or all of mathematics to logic as it was an extension of logic to include (parts of) mathematics.5 This observation raises an immediate question which a logicist must address: How to justify the claim that the discipline to which mathematics is reduced really is logic, rather than something else? This complaint has often been directed against logicism.6 An early expression to it was given by Henri Poincaré. In one of his highly critical surveys of the new logic he puts the point as follows:

We see how much richer this new logic is than the classical logic. The symbols have been multiplied and admit of varied combinations, which are no longer of limited number. Have we any right to give this extension to the meaning of the word logic? It would be idle to examine this

4 Russell (1901a, p. 367). 5 Cf. Tiles (1980, p. 158). 6 The complaint was accentuated by the discovery of paradoxes and the complications it necessitated in the “logic” underlying the reductionist ambitions. Indeed, it seems that Frege’s reason for giving up logicism was that he found himself unable to regard the ensuing complications as a matter of logic. Russell was in this respect much more flexible, and at least in the case of the theory of types he managed to convince himself that the type- theoretical hierarchy was really “plain common sense” (1924, p. 334; Lakatos (1962, pp. 12-8) has some very good remarks on this issue as it applies to Russell). However, the complaint has been there from the beginning of modern logic. Ch.1 Preliminary Remarks on Russell’s Early Logicism 29

question and quarrel with Mr. Russell merely on the score of words. We will grant him what he asks; but we must not be surprised if we find that certain truths which had been declared to be irreducible to logic, in the old sense of the word, have become reducible to logic, in its new sense which is quite different. (Poincaré 1908, p. 162)

Russell’s logicism, Poincaré seems to be saying, involves no more than a change in the way the word “logic” is used. In this way, there really is no clash between the two logics, traditional and Russell’s, for the simple reason that tradition and Russell are talking about different things. Whether or not this really is what Poincaré had in mind,7 at least we can see him as making the following unquestionably valid point: in the absence of a further characterisation of logic, a logicist or any other advocate of the new logic wouldn’t be entitled to draw distinctively philosophical consequences from his results. But, of course, logicists typically did draw just such philosophical consequences from their reductive theses. For example, Russell argued that Kant had been wrong about the nature of mathematical reasoning and that, when properly construed, these reasonings do not presuppose an extra-logical source. It is clear that when Russell argued for these views, he did not consider himself redefining the term “logic”. Rather, he saw himself as revealing what is really involved in deductive reasoning in general and mathematical reasoning in particular. It is this “what is really involved” that underlies Russell’s criticism of Kant, and at least from Russell’s point of view the disagreement between himself and Kant was a genuine one, rather than just a matter of words.

7 It is in fact clear that Poincaré is not just endorsing a change of meaning-thesis. For he argues that the logicist use of “undemonstrable principles” involves appeal to intuition. Hence these principles are mathematical, rather than logical, according to him; referring to these principles, he writes: “Have they altered in character because the meaning of the word has been extended, and we find them now in a book entitled ‘Treatise on Logic’? They have not changed in nature, but only in position” (1908, p. 162). 30 Ch.1 Preliminary Remarks on Russell’s Early Logicism

In order, therefore, to gain understanding of Russell’s conception of logic, we can turn, first, to the content of his early logicism and, second, to the philosophical lessons that he derived from it. Evi- dently, the philosophical consequences that one takes logicism to have are determined by one’s conception of logic. The main interpretative question that I shall address in this work is therefore this: What must logic be like in order for the logicist reduction to have the importance that Russell thought it had? I shall begin to answer this question by considering, in this chapter, two questions about Russell’s early version of logicism. Firstly, I will examine what relation it bears to the Fregean and logical empiricist versions of logicism. Secondly, I will consider the question: How did the early Russell understand the point behind his logicism? An answer to this question forms an examination of the mathematical background of his logicism.

1.1 Different Logicisms

It is generally recognised that Russell saw Kant’s theory of mathematics as the main target of his own logicist philosophy of mathematics. This is what he himself said when looking back on his philosophical career.8 There is also ample textual evidence for this in the relevant writings from the early logicist period. There we find more than one stricture on the “Kantian edifice”, which, Russell argued, had been torn down by modern mathematics and logic.9 What is less clear is precisely how Russell thought his logicism would contribute to the collapse of Kantianism. We can begin the

8 “The primary aim of Principia Mathematica was to show that all pure mathematics follows from purely logical premisses and uses only concepts definable in logical terms. This was, of course, an antithesis to the doctrines of Kant, and initially I thought of the work as a parenthesis in the refutation of ‘yonder sophistical philistine’, as Georg Cantor described him” (Russell 1959, pp. 74-5; cf. also Russell 1944, p. 13). 9 I shall give several quotations below. Ch.1 Preliminary Remarks on Russell’s Early Logicism 31 clarification of this interpretative question by relating Russell’s early logicism to two other philosophies of mathematics known by the name of “logicism”, namely Frege’s thesis concerning the nature of arithmetic and the conception of mathematics that logical empiricists developed in the 1920s and 30s. Both Frege and Russell and the logical empiricists thought it was philosophically enlightening to relate their own views on mathematics to Kant’s. Indeed, it is quite common to subsume all these three logicisms under one and the same label and also to formulate their (or its) philosophical point in terms of a consciously held opposition to Kant.10 Closer inspection shows, however, that there were as many logicisms as there were logicists. What may cause confusion and prevent us from seeing and appreciating the divergent purposes of different logicists is the fact that the philosophical import of logicism is commonly described with the help of two familiar distinctions: analytic vs. synthetic and apriori vs. aposteriori.11 Thus, a more or less standard characterisation of logicism would attribute the following two points to it: firstly, the logicist reduction shows that mathematics, being reducible to logic, is analytic and (for that reason) apriori; secondly, this shows that Kant

10 It should be noted that the term “logicism” (or its German equivalent, “Logizismus”) was used neither by Russell nor Frege. It was only in the late 1920s that “Logizismus” was used, by Fraenkel and Carnap, to denote a certain position in the philosophy of mathematics (this according to Grat- tan-Guinness (2000, pp. 479, 501)). In his (1931) Carnap draws the once popular three-fold distinction between logicism, intuitionism and formalism. Concerning the first, Carnap writes: “Logicism is the thesis that mathematics is reducible to logic, hence nothing but a part of logic. Frege was the first to espouse this view (1884). In their great work, Principia Mathematica, the English mathematicians A. N. Whitehead and B. Russell produced a systematization of logic from which they constructed mathematics” (id., p. 31). 11 Instead of the Latin phrases “a priori” and “a posteriori” I use “apriori” and “aposteriori” as single English words; in this I follow Burge (1998a, p. 11, fn.). In quotations I use whichever phrase is used by the author. Also, I use “apriority”, rather than “aprioricity”, which is “barbaric”, according to Burge. 32 Ch.1 Preliminary Remarks on Russell’s Early Logicism was mistaken in his view that mathematical truths are synthetic and apriori.

1.2 Analyticity and Syntheticity

1.2.1 Preliminary Remarks

It is not difficult to find passages from the relevant authors which appear at first sight to lend support to the attribution of this pair of views to logicists. As for Russell, one might refer to My Mental Devel- opment, a short intellectual autobiography written for the Schilpp- volume dedicated to his own philosophy. There we find him mentioning that he “did not like the synthetic a priori”, which was the reason why he “found Kant unsatisfactory” in the philosophy of mathematics (1944, p. 12). Similarly, Frege explains in his Grundlagen der Arithmetik that the goal of the work is to make probable the view that the truths of arithmetic are analytic and apriori, a view that he saw as a correction of Kant (Frege 1884, §§88-9). As for logical empiricism, one could refer to practically any author associated with it, but A. J. Ayer’s Language, Truth and Logic gives a particularly powerful expression to one of their leading ideas, namely that the analyticity of mathematical truth explains its necessity and apriority, and thereby enables one decisively to undermine a particularly annoying case of the Kantian synthetic apriori (Ayer 1936, ch. 4).12 Given passages like

12 To speak of the logical empiricists’ views on mathematics as one logicism is to commit oneself to an oversimplification which is itself yet another instance of the one that is discussed in the text. Ayer, for instance, was not very much interested in logicism and its prospects as in maintaining the analyticity (and hence apriority) of mathematics. Indeed, the truth or otherwise of logicism is of less importance to him, since he claims that on his conception of analyticity – the criterion of the analyticity of a proposition being that “its validity should follow simply from the definition of the terms contained in it” (1936, p. 109) – the propositions of pure mathematics are analytic whether or not pure mathematics is reducible to logic in Russell’s Ch.1 Preliminary Remarks on Russell’s Early Logicism 33 these, the conclusion lies in hand that there was a philosophy of mathematics, appropriately called “logicism”, the point of which was to show that since mathematics is reducible to logic, it is analytic and apriori, contrary to what some, Kant in particular, had thought. That the term “logicism” lends itself to this formulation, however, serves only to hide the fact, disclosed by a closer look at the relevant authors, that there is not just one logicism, but, indeed, three distinct theories, different from one another in motivation, content and scope.13

1.2.2 Kantian Analyticity

For Kant, the importance of the distinction between analytic and synthetic was primarily epistemological. He had made the observation that there is a class of judgments which are not only knowable apriori but whose apriority is unproblematic, namely, those that he called “analytic”. Informed by what generations of philosophers and logicians had taught about concepts and judgments, he gave the following preliminary characterisation of analyticity: a judgment is analytic if “the predicate B belongs to the subject A, as something that is (covertly) contained in this concept A” (Kant 1781/1787, A6/B10).14 As the quotation shows, conceptual containment comes in two kinds: explicit, as in “all amphibious animals are animals”, and implicit, as in sense (ibid.) The standard characterisation of the logical empiricist appropriation of logicism best fits someone like Carnap, who, by suitably extending and modifying the Tractarian notion of tautology, arrived at the conception that “all valid statements of mathematics are analytic in the specific sense that they hold in all possible cases and therefore do not have any factual content” (1963, p. 47; for details, see Friedman 1997). 13 There are other philosophies of mathematics which can be classified as logicisms and which differ from those discussed in the text. An example would be Dedekind’s views on arithmetic in his (1888), where a version of logicism is explicitly affirmed. 14 Kant’s first Critique will be referred to in the standard manner, by citing the relevant page or pages of the A- (1781) and/or B- (1787) editions. 34 Ch.1 Preliminary Remarks on Russell’s Early Logicism

“all bachelors are men”. A judgment of the second kind is one which adds “nothing through the predicate to the concept of the subject, but merely [breaks] it up into those constituent concepts that have all along been thought in it, although confusedly” (A7/B11). This explanation leads to a second characterisation of analyticity, one that is based, in effect, on the sort of proof that is appropriate for analytical truths: since every analytical truth is either explicitly or implicitly of the form “(every) AB is B”, our knowledge of an analytical truth is either immediate (“all amphibious animals are animals”) or mediated by a proof of the sort which leads from “all bachelors are men” via a simple substitution to “all unmarried men are men”. Thus, according to Kant, all analytical truths are grounded in the principle of contradiction, which is at the same time the principle sufficient for analytical knowledge (A150-151/B190-191). It is this feature of analytical truths which makes them epistemologically unproblematic: there is no problem about their content being apriori knowable; we need not consult experience in order to come to possess the piece of knowledge î if, indeed, it deserves to be called knowledge î that all bachelors are unmarried. We cannot fail to recognise the truth of this judgment if only we possess the relevant concepts.15 Kant then points out that analytical judgments by no means exhaust the class of truths that are knowable apriori. For reflection on the relevant instances shows that there is a further class of judgments which shares with the first one the property of being apriori but has the additional property of being epistemologically problem-

15 To make this explanation complete and convincing, one would have to explain the nature of this “cannot fail to recognise”. Perhaps we should say that it is a criterion for possessing the concept of bachelor that one assents to or accepts as true the judgment that all bachelors are men. This formulation would explain in a non ad hoc manner why there can be no gap between possessing a concept and recognising a certain truth. We shall in fact return to this model of explanation repeatedly, when we consider different “models” for apriori knowledge: see sections 2.7, 3.6.3, 5.5 and 5.10. Ch.1 Preliminary Remarks on Russell’s Early Logicism 35 atic in the sense that no straightforward explanation of apriority is forthcoming in this case.16 For Kant, then, the importance of the analytic-synthetic distinction is in the first instance epistemological: in particular, the notion of analyticity was intended by him to single out a class of judgments for which mere analysis of their content yields as a corollary an explanation of their epistemology, i.e., their being apriori knowable. When we turn to logicists, it is precisely on this point that we can discern important differences.

1.2.3 Logical Empiricists and Analyticity

Consider first logical empiricists. In the use to which they put the notion of analyticity they were followers of Kant. In both cases the importance of analyticity stems from the fact that it is intended to play an explanatory role. In this sense the logical empiricist notion of analyticity was, as Paul Benacerraf has put it, “an extension of Kant’s distinction and of the epistemological analysis that went along with it” (1981, p. 53).17 That a truth is analytic in the logical empiricist sense – true solely in virtue of meaning and hence true, somehow, in virtue of linguistic rules and, for that very reason, devoid of what was known as “factual content” – is meant to explain why it is knowable independently of experience in precisely the way that Kant meant his notion of analyticity to explain the apriority of “merely explicative” truths. Of course, the logical empiricist path from analyticity to apriority is much more difficult than the Kantian one, since the former comprises much more than those trifling truths that can be

16 In J. A. Coffa’s succinct formulation: “[i]n pre-Kantian philosophy, many had assumed that the notion of analyticity provided the key to apriority. Kant saw that a different account was needed since not every a priori judgment is analytic, and offered a new theory based on one of the most remarkable philosophical ideas ever produced: his Copernican turn” (1991, p. 2). 17 See also Skorupski (1993, 1995). 36 Ch.1 Preliminary Remarks on Russell’s Early Logicism covered by the Kantian conceptual containment model for analytical judgments. Having in this way extended the scope of analytical truths, logical empiricists found themselves in a position to argue for a decidedly anti-Kantian conclusion: nothing beyond a grasp of the rules of a relevant language is needed to explain why mathematical truth is knowable independently of experience. And this was meant as a partial answer to Kant’s epistemological question about the source of our apriori knowledge.18 According to logical empiricists, there is no need to assume with Kant a distinctive type of truth – that which is synthetic and yet knowable apriori – nor to postulate a special source of knowledge (intuition) to guarantee access to these truth. In spite of their reaching radically different conclusions, the point of introducing analyticity is precisely the same in the two cases: to explain how the truth of a statement can be recognised without having to rely on empirical input, that is, merely by entertaining its conceptual content.

1.2.4 Analyticity in Frege and Russell

Consider next the case of Frege. Although deceptively similar, his use of analyticity and his logicism must be distinguished from the logical empiricist versions. Although Frege has much to say about analyticity – reducibility to logic via explicit definitions – he has, apparently anyway, next to nothing to say about our knowledge of logic.19 It is true that he claims that his intention is not to assign any new meaning to the terms “analytic” and “synthetic” but that he only wants to state precisely what earlier authors, Kant in particular, had had in mind (1884, §3). Yet, it is no less true that Frege was largely insensitive to

18 “Partial” since it has to be complemented by an account of the nature of language and our knowledge of it that can be deemed acceptable by the logical empiricists’ standards. But here they could refer to conventionalism and behaviourism. 19 See, however, section 5.5 for a more accurate description of Frege’s situation. Ch.1 Preliminary Remarks on Russell’s Early Logicism 37 the epistemological problems that had exercised Kant and were to exercise logical empiricists.20 As far as Frege is concerned, the crucial point can be put briefly by saying that he was – most of the time, anyway – content with the assumption that there is knowledge that is apriori, without bothering to provide an explanation of what this knowledge is like.21 Consider finally Russell’s early logicism. There are, of course, conspicuous differences between Russell’s and Frege’s logicisms. Nevertheless, when it comes to the issue of analyticity, their views bear an important similarity to one another. Above all, as in Frege’s case, the motives that Russell had for his logicism must be distanced from the epistemological underpinnings of logical empiricism. As several scholars have pointed out, Russell has in fact very little use for the notion of analyticity.22 The notion of analytic truth, or the distinction between analyticity and syntheticity, is afforded absolutely no role in the Principles. It receives no extended discussion and is not put into any use. Indeed, he almost fails to mention it, and when he once mentions it, he does so only to put it aside as being of no concern to him. Moreover, what little he says seems to distinguish him firmly from logical empiricism. This is what Russell has to say about the notion:

Kant never doubted for a moment that the propositions of logic are analytic, whereas he rightly perceived that those of mathematics are synthetic. It has since appeared that logic is just as synthetic as all other kinds of truths: but this is a purely philosophical question, which I shall here pass by. (Russell 1903a, §434)

Taking this passage into account prevents us from applying the standard characterisation of logicism to the early Russell: if logic,

20 See, for example, Skorupski (1984, pp. 239-40). 21 See Skorupski (1993, pp. 142-3; 1995). This view may be a little too simple as a complete characterisation of Frege’s views (see, again, section 5.5), but it will do for now. 22 See Taylor (1981), Coffa (1982), Hylton (1990a, p. 197; 1990b, p. 204). 38 Ch.1 Preliminary Remarks on Russell’s Early Logicism according to Russell, has turned out to be synthetic, we can hardly say that the point of his logicism was to show that mathematics is analytic. On the other hand, saying it is synthetic is something that seems to have little interest for him. In thus rejecting the terminology of “analytic” and “synthetic”, Russell also seems to be deviating from Frege. However, it is worth pointing out – and this is something that has not been widely recognised – that Russell was ready to express agreement with Louis Couturat when the latter proposed a definition of analyticity that was exactly like the one that Frege had given in Grundlagen:

We may usefully define as analytic those propositions which are deducible from the laws of logic; and this definition is conformable in spirit, though not in the letter, to the pre-Kantian usage. Certainly Kant, in urging that pure mathematics consists of synthetic propositions, was urging, among other things, that pure mathematics cannot be deduced from the laws of logic alone. In this we now know that he was wrong and Leibniz was right: to call pure mathematics analytic is therefore an appropriate way of marking dissent from Kant on this point (Russell 1905a, p. 516; italics added).

In the end, then, Russell did not object to applying the notion of analyticity to “pure mathematics”. But as we saw above, this sense which Russell was ready to accept – the Fregean one – is in a crucial respect different from what Kant and logical empiricists had in mind. The following quotation from Warren Goldfarb gives a succinct formulation of this conclusion: “In fact, no real role is played by any distinction between analytic and synthetic in early logicism. The central and basic distinction for both authors [sc. Frege and Russell] is that between logical truth and extralogical truth. The question, then, is that of discerning those features of the new logic which enabled it to work so effectively against Kant” (1982, p. 693). The point is this. Even though it is a trivial consequence of Frege’s definition of analyticity that logic is analytic, the definition does not as such deliver any further characterisation of logic that would explain the importance of the purported reduction. Ch.1 Preliminary Remarks on Russell’s Early Logicism 39

When it comes to analyticity and syntheticity, we must distinguish Frege’s and Russell’s use of these concepts from Kant, for whom analyticity is spelled out by dint of the notion of conceptual containment, as well as from logical empiricists, who resorted to “truth in virtue of meaning”. This difference can be expressed as follows. Neither the notion of conceptual containment nor that of truth in virtue of meaning contain reference to logic. It follows that for Kant and the logical empiricists it is, as it were, a further discovery that the discipline one calls “logic” is one to which analyticity applies, whereas Frege’s – and, on occasion, Russell’s – procedure is the exact opposite. For this reason Frege and Russell cannot use analyticity for any explanatory purposes (and do not intend so to use it). As Goldfarb says, for them the crucial distinction is that between logical and extra- logical truth. This point is seen clearly by reflecting on the above quotation from Russell. He did not dispute Kant’s claim that pure mathematics is not “deducible from the laws of logic”, if logic is understood in the way Kant understood it. If he had disputed this, there would have been no need for reform in logic. Hence the slogan “pure mathematics is reducible to logic” is “an appropriate way of marking dissent from Kant” only when it is conjoined with an articulation of what distinguishes logical truth from non-logical truth and how this undermines Kant’s theory of mathematics.23 This being said, we can conclude that the analytic-synthetic distinction had no important role to play in the early Russell’s logicism.

23 We can now see what Russell meant when he wrote in the Principles of Mathematics that “[i]t has since appeared [sc. after Kant’s days] that logic is just as synthetic as all other kinds of truth”. This means simply that the new logic of Peano, Russell and others was not a body of trifling truths and principles – a view that Kant had applied to the formal logic of his time – and cannot therefore be classified as analytic in Kant’s sense. Russell’s claim that logic is synthetic is not an exciting philosophical thesis but a recognition of what he took to be an undeniable fact. This, however, did not lead Russell to reflect on the epistemological status of this new logic, which shows him to have been an ally of Frege in this respect and distinguishes him clearly from Kant and logical empiricists. 40 Ch.1 Preliminary Remarks on Russell’s Early Logicism

Certainly it had no role comparable to that given to it in logical empiricism. If we draw the distinction within the context of Russell’s logicism at all (and as we have seen, this can certainly be done), then it is essentially identical with Fregean analyticity. This notion, however, does not ascribe to logic any further characteristic which would turn “analyticity” into an explanatory concept. For this reason Rus- sell’s logicism, like Frege’s, must be kept firmly distinct from the later use to which their ideas were put in the hands of someone like Car- nap or Ayer. This means in particular that Russell’s reasons for pursuing logicism cannot be brought to the fore simply by referring to the analytic-synthetic distinction. It comes as no surprise to hear that Frege’s and Russell’s appropriation of “logicism” was significantly different from the logical empiricist one. Their opposition to empiricism was indeed quite fundamental and stems from a firm conviction that consistent empiricism is inconsistent with a viable philosophy of mathematics and logic. First and foremost, they regarded empiricism as being irreme- diably involved in psychologism, and this was for them a sufficient reason for dismissing it as a confused piece of philosophising.24

24 Frege’s opposition to empiricism is evident, for example, from his dismissal of Mill’s attempt to ground arithmetical definitions in observed matters of fact. Frege admits that the idea of grounding a science in definitions is sound; nevertheless, Mill’s execution of this idea is flawed “thanks to his preconception that all knowledge is empirical” (1884, §7). It can be seen, then, that Frege’s criticism of Mill was not that the latter’s theory was psychologistic. Yet, Frege thought that at least at the level of logic the Millian preconception results in psychologism with its characteristic confusion of what is objective with what is subjective (the subject-matter of logic, or the science of the laws of thought, consists in mental items and their interrelations). Russell is in this respect more sweeping and sees in empiricism an immediate commitment to psychologism: “[m]isled by neglect of being, people have supposed that what does not exist is nothing. Seeing that numbers, relations, and many other objects of thought, do not exist outside the mind, they have supposed that the thoughts in which we think of these entities actually create their own objects” (1903, §427). Thus he dismisses the Millian account of numbers offhand and contends that the failure to Ch.1 Preliminary Remarks on Russell’s Early Logicism 41

Frege’s anti-psychologism is both well-known and extensively discussed in the secondary literature.25 It was also of crucial significance for Russell. He had published his own (short) criticism of psychologism as early as 1895 (Russell 1895, pp. 251-2), while he was still an idealist. In this he was merely following such better-known idealists as F. H. Bradley, whose verdict on empiricist associationism and the resulting psychologism was not only as harsh as Frege’s but also bears important similarities to it.26 While no longer an idealist, the logicist Russell remained every bit as hostile to psychologism. It is true that he now associated psychologism with idealism, rather than empiricism.27 This, however, does not signal any change in his attitude towards empiricism; rather, it has to do with the fact that there were not many people in Russell’s intellectual environment at the turn of the century for whom empiricism would have been a live option.28 observe the distinction between being and existence leads immediately to psychologism. Though empiricism is not mentioned here by name, it is clear that Russell’s remark is intended to apply to it as well. According to Moore’s and Russell’s new realism, the characteristic thesis of empiricism, to wit, that “experience is the origin of all knowledge”, amounts to the view that “all known truths are truths about what exists at one or more moments of time” (Moore 1902-3, pp. 91-3), thus implying precisely, a failure to observe the distinction between being and existence. 25 See, for example, Notturno (1985) or Baker and Hacker (1989). 26 See Gerrard (1997, sec. VII) f or a discussion of Frege, Bradley, Russell (and Moore) on anti-psychologism. 27 Russell’s criticism of Kant’s transcendental idealism is a prominent example of how idealism was thought to get entangled in psychologism. See below, section 3.6.2.1. 28 It seems also clear that Frege and the early Russell would have seen little reason to revise their appreciation of empiricism even if they had been acquainted with it in some more refined form than that exhibited by Mill; if, for instance, they had been familiar with logical empiricists, who were particularly concerned with bringing the Millian preconception into harmony with the existence of such putatively apriori disciplines as mathematics and logic. Logical empiricists were quite as much opposed to psychologism as were Frege and Russell, so that the latter would have been forced to refine their criticisms. Nevertheless, there is little reason to think that they would 42 Ch.1 Preliminary Remarks on Russell’s Early Logicism

1.3 The Pursuit of Rigour

1.3.1 The Mathematical Context

The considerations presented above show that Russell’s (and Frege’s) logicism must be kept separate from theses that are characteristic of logical empiricism. If one wants to have a catchword to describe the philosophical import of these earlier logicists, there is a much better one in the offing than analyticity, namely rigour. This term carries with it a suggestion about both the proper historical context within which these earlier logicists should be located and the point behind the logicist reduction. As regards context, “rigour” enjoins us to see such early logicists as Frege and Russell not as originators of certain conception characteristic of 20th century philosophy but, rather, as standing close to a tradition that was essentially mathematical. As regards the point of logicism, it must be admitted that Russell’s (and Frege’s) repeated insistence on the necessity of formulating mathematical theories rigorously is not self-explanatory, as this makes explicit neither the content of rigour nor the point behind it. Nor can this issue be clarified simply by referring to the mathematical background, the tradition of “establishing mathematics rigorously” (Kline 1990, p. 1025), since that mathematical tradition itself as well as its precise relation to the early logicists raises its own questions of interpretation. By unravelling at least some of these questions, as they relate to have found the logical empiricist conception of apriority as analyticity and its implications as less misguided than the somewhat crude psychologism which was characteristic of 19th century naturalism and empiricism. Insofar as the logical empiricist notion of analyticity involves an account of logical validity (or logical truth or logical necessity) which traces the relevant notion back to our accepting one particular set of linguistic rules î as Carnap suggests with his famous words, “[e]veryone is at liberty to build up his own logic (i.e. his own form of language) as he wishes” (1934, p. 52) î they would have regarded the ensuing logical tolerance, if spelled out, as a reductio of the proposed account (whether this charge is justified and how the convention- alist position could be defended are interesting issues which will not be pursued here). Ch.1 Preliminary Remarks on Russell’s Early Logicism 43

Russell’s logicism, we gain a better understanding of the precise import (both positive and negative or critical) of his early logicism as and the proper historical and philosophical context in which it should be located. According to the present interpretive suggestion, then, the early Russell’s logicism should be considered in the context of certain developments which had taken place in mathematics in the 19th century. These developments resulted in a transformation readily comparable to that which occurred in logic; in both cases one is justified in speaking about “a transformation so profound that it is not too much to call it the second birth of the subject” (Stein 1988, p. 238). As for mathematics, the development has two sides to it. Firstly, mathematicians extended their research to new areas, which resulted in the discovery or invention of entirely new kinds of mathematical objects with methods to deal with them. Thus, the 19th century saw the introduction of such new areas as projective and non-Euclidean geometry, set theory and transfinite mathematics, complex and imaginary numbers, infinitely distant points and points with imaginary co-ordinates and algebraic numbers – even though in some cases, like projective and non-Euclidean geometry, the beginnings had been laid by earlier mathematicians. The second characteristic feature are the profound changes to which existing branches of mathematics were subjected. The standard example is the evolution of modern analysis from Newtonian and Leibnizian Calculus through the efforts of mathematicians like the Bernoullis, Euler, Bolzano, Cauchy, Abel, Dirichlet and Riemann into the definitive form given to it by Weierstrass, Dedekind and Cantor. It is this second feature above all which has given rise to such terms as “pursuit of rigour” and “revolution in rigour”. Applied to the history of the calculus, this terminology refers to the characteristically nineteenth century achievement of giving analysis the form in which it is still known and practised, an accomplishment known as the “arithmetisation” or, indeed, “rigorisation” of analysis. It resulted in a thoroughgoing change in the accepted concepts and methods of the calculus; as Morris Kline has put it, 19th century mathematicians 44 Ch.1 Preliminary Remarks on Russell’s Early Logicism

“freed the calculus and its extensions from all dependence upon geometrical notions, motion and intuitive understandings” (1990, p. 972). The notion of rigour, however, is by no means restricted to the development of the calculus. Of particular importance (also for Russell) was the emergence of non-Euclidean geometries (Gauss, Bolyai, Lobachevski, Riemann, Beltrami), which had at least two implications for anyone with a philosophical interest in mathematics. Firstly, it forced mathematicians to reconsider hitherto accepted proofs and definitions and the very principles underlying mathematical practices. Kline writes that the effects non-Euclidean geometries had on mathematicians were “disturbing” and describes them as follows: “[n]ot only did this [sc. the creation of non-Euclidean geometries] destroy the very notion of the self-evidency of axioms and their too-superficial acceptance, but the work revealed inadequacies in proofs that had been regarded as the soundest in all of mathematics. The mathematicians realized that they had been gullible and had relied on intuition” (id., p. 1025). Secondly, the creation of non- Euclidean geometries challenged some deep-rooted conceptions as to the nature of mathematical knowledge (cf. here Detlefsen 1995). When mathematicians established that non-Euclidean geometries were consistent (relative to Euclidean geometry), it began to seem as if there was a crucial difference between arithmetical and geometrical knowledge; the former is, indeed, apriori, but one cannot decide between Euclidean and non-Euclidean geometries on apriori grounds (this was the conclusion that Gauss, for instance, drew). Therefore, a philosophically minded mathematician or a mathematically alert philosopher had to find a way to explain the difference between arithmetic and geometry. The term “rigour” is commonly used with the intention of bringing out a sharp break of nineteenth century mathematicians with their predecessors, a break which occurred when accepted mathematical practices were subjected to a critical scrutiny and reappraisal. Al- though there is room for dispute about what brought about this reappraisal, the transformation itself is an undeniable fact about the history of mathematics. Similarly unproblematic is the broad interpre- Ch.1 Preliminary Remarks on Russell’s Early Logicism 45 tative suggestion that Russell’s logicism should be embedded in the context created by this transformation; his logicist philosophy of mathematics was clearly an outgrowth of, or a philosopher- mathematician’s reaction to, such results as the rigorisation of analysis, the creation of non-Euclidean geometries, Cantor’s set-theory and Peano’s mathematical logic, all of which contributed to the transformation of mathematics in one or other of the two senses distinguished above.

1.3.2 Why Rigour?

This brief charting of the background of Russell’s logicism does not as yet explain the sense that he attached to rigour. As was pointed out above, “rigour” is not a self-explanatory term. In order correctly to understand the point behind Russell’s logicism, we must acknowledge that the term is ambiguous in a way that relates to the very point and purpose of such foundational programmes as logicism.29 The terminology of rigour involves an important and often unac- knowledged ambiguity which relates to the very point and purpose of such foundational programs as logicism. In one sense, rigour is an epistemological notion. In this sense, the search for a firm or rigorous foundation is a search for an epistemically secure or unshakeable ground. In the second sense, the pursuit of rigour is a broadly semantic enterprise that relates to the improvement of mathematical understanding, rather than mathematical knowledge. It is somewhat less easy to give a brief characterisation of this second sense than it is to describe the importance that an epistemically secure foundation might have for someone. Following Coffa (1991, p. 27), we may say that this second sense involves “a search for a clear account of the basic notions of a discipline”. We can see the semantic sense of rigour in use when we consider Russell’s view that rigorous definitions of such notions as infinity,

29 The following discussion draws on Coffa (1991, pp. 26-8). 46 Ch.1 Preliminary Remarks on Russell’s Early Logicism continuity and limit had made it possible to form a correct conception of such philosophically pregnant topics as space, time, motion and change.30 Given the mathematical results established by Weier- strass, Cantor and Dedekind, earlier philosophers’ theories of these matters (and here Russell has Kant and other idealists in mind) were capable of being conclusively refuted for the simple reason that they had not really known what they were talking about. To illustrate, consider the case of infinity. If it is accepted as an evident truth that a whole is always greater than a part (as many philosophers had done), then the infinite – and anything in which it is involved – must be condemned and rejected as contradictory. Where philosophers went wrong, Russell explains, is that they never asked what infinity is, and their actual treatments of the topic show that had they ever been asked the crucial question, they “might have produced some unintelligible rigmarole, but [they] would certainly not have been able to give a definition that had any meaning at all” (1901a, p. 372). Mathemati- cians, by contrast, not only posed this question to themselves but also gave “a perfectly precise definition of infinite number or an infinite collection of things” (ibid.) It follows from these developments that modern mathematics had deprived philosophers’ views on infinity and related notions of whatever plausibility they may have possessed before mathematicians gave their exact definitions and notions with perfectly precise meanings. Rigour is important, furthermore, not only because it enhances our understanding of such topic as infinity. Even more important for Russell was the fact that mathematicians pursuing rigour had in fact changed the whole of their science: “one of the chief triumphs of modern mathematics”, he wrote, “consists in having discovered what mathematics really is” (1901a, p. 366). From a philosophical point of view, then, the pursuit of rigour put a philosopher of mathematics for

30 This charge is stock-in-trade of Russell’s. For an early formulation see Russell (1901a, pp. 372-3). Elsewhere he goes so far as to say that “[t]he solution of the difficulties which formerly surrounded the mathematical infinite is probably the greatest achievement of which our age has to boast” (1907a, p. 66). Ch.1 Preliminary Remarks on Russell’s Early Logicism 47 the first time in a position to give a non-arbitrary answer to the question of the nature of mathematics. Russell’s logicism is an attempt to give a detailed answer to this question. Even though the semantic sense is evidently present in Russell’s writings – and is arguably what gives his logicism its real bite, a claim which will be developed in detail below – later interpreters have in general tended to interpret his interest in the foundations of mathematics in the light of what Coffa calls the epistemological sense of rigour. According to a common conception, Russell’s logicism was a special case of his more general preoccupation with such issues as scepticism, certainty and the possibility of knowledge; on this view Russell wanted to find good reasons to believe that mathematics gives us genuine – that is, certain, indubitable – knowledge.31 It was for this reason, it is argued, that he attempted to demonstrate that mathematics is just logic: if that turns out to be the case, then we can be assured that there is certainty and genuine knowledge.32

31 For this view, see Andersson (1994). 32 This very same ambiguity affects also one’s reading of Frege. He was clearly concerned with rigour in the semantic sense when he laments, in the Preface to Grundlagen, arithmeticians’ ignorance of number one (that is their inability to give an acceptable answer to the question, what number one is), and says that, consequently, we “hardly succeed in finally clearing up negative numbers, or fractional or complex numbers, so long as our insight into the foundation of the whole structure of arithmetic is still defective” (1884, p. ii). In order to rectify the situation, he thought, it was necessary to provide arithmetic with genuine foundations, which would, among other things, tell us what the number one is. Nevertheless, Frege’s logicism is often interpreted in a way that connects his insistence on rigour with skepticism and a search of certainty. This reading of Frege is found, for instance, in Currie (1982, pp. 28-9). 48 Ch.1 Preliminary Remarks on Russell’s Early Logicism

1.3.3 Epistemic Logicism

Epistemic logicism is any variant of logicism that seeks to derive a primarily epistemological lesson from the logicist reduction.33 For an epistemic logicist, the fundamental problem is the justification, or explanation, of mathematical knowledge, and his aim is to account for this by reducing the epistemology of mathematics to that of logic. On this view, the importance of the logicist reduction stems from the fact that it claims to reveal the true ground of justification for mathematical propositions. Because we are epistemically justified in accepting the basic truth of logic î say, because they are self-evident î the identification of mathematics with logic would extend this justification from logic to mathematics.34 The principal source for the interpretation which associates Russell’s work on the foundations of mathematics with epistemic logicism is the commonly held conception of him as a philosopher for whom the overriding question of philosophy was one concerning knowledge and certainty.35 And it must be said that it is Russell himself who is chiefly responsible for this interpretation. On the face of it, epistemic logicism is a direct consequence of this “constant preoccupation” of his (Russell 1959, p. 11). The following quotation, in which he speaks of his work leading up to Principia Mathematica, is typical:

The desire to discover some really certain knowledge inspired all my work up to the age of thirty-eight. It seemed clear that mathematics had a better claim to be considered knowledge than anything else; therefore it was to the principles of mathematics that I addressed myself. At the age of thirty-eight I felt that I had done all that lay in my power to do in this field, although I was far from having arrived any absolute certainty. Indeed the net result of my work was to throw doubt upon arithmetic which had never been thrown before. I was and am persuaded that the method I pursued brings one nearer to knowledge than any other that is

33 The term “epistemic logicism” is taken from Irvine (1989, sec. 2). 34 Irvine (1989, p. 307). 35 See Wood (1959). Ch.1 Preliminary Remarks on Russell’s Early Logicism 49

available, but the knowledge it brings is only probable, and not so precise as it appears at first sight (Russell 1929, p. 16; for a very similar statement, see Russell 1924, pp. 323-4).

The following quotation is even more to the point, since it forges an explicit connection between one important aspect of the concept of rigour, that of rigorous proof, and its potential epistemic virtues:

I wanted certainty in the kind of way in which people want religious faith. I thought that certainty is more likely to be found in mathematics than elsewhere. But I discovered that many mathematical demonstrations, which my teachers expected me to accept, were full of fallacies, and that, if certainty were indeed discoverable in mathematics, it would be in a new kind of mathematics, with more solid foundations than those that had hitherto been thought secured (1956a, p. 53).36

Imre Lakatos’ paper “Infinite regress and foundations of mathematics” (Lakatos 1962) constitutes what is probably the most striking attempt to follow Russell’s own testimonies and make his logicism a variant of epistemic logicism.37 According to Lakatos, logicism should be looked upon as an attempt to counter the classical sceptical argument from infinite regress. As a sceptic uses it, this argument is meant to show that any claim to knowledge can be undermined since no firm foundation can be given either for meaning or for truth; every effort at establishing either of these can always be met with a sceptical demur, and insofar as one tries to meet this demur by giving the meaning of an expression in terms of other expressions or reducing the truth of a statement to that of other statements, the sceptic can always reply by reasserting his position. Not even the claim that one has reached an absolute end-point – meanings that are perfectly

36 The passage is repeated in Russell (1969), p. 220. 37 According to Lakatos, his interpretation applies to Frege as well, although he uses Russell’s logicism as an illustration. In fact, Lakatos goes so far as to say that he wants to “exhibit modern mathematical philosophy as deeply embedded in general epistemology and as only to be understood in this context” (1962, p. 4). 50 Ch.1 Preliminary Remarks on Russell’s Early Logicism transparent or truths that are absolutely self-evident – can defeat the sceptic, since the requirement for a warrant applies with no less force to them. The problem, as Lakatos puts it, is that “[m]eaning and truth can be transferred, but not established” (1962, p. 3). Lakatos then suggests that Russell’s logicism is an instance of what he calls the “Euclidean programme” (id., p. 4). That is, it is in essence an endeavour to save mathematics from sceptical doubt that is kept alive by the constant threat of the infinite regress. This it attempts to do by showing, in accordance with epistemic logicism, how mathematics can be derived from self-evident logical axioms and abbreviatory definitions by means of self-evidently correct rules of inference. On Lakatos’ reading, the rationale for pursuing the logicist reduction stems from the inner instability of the Euclidean programme; concerning any putative foundation for (mathematical) knowledge, the sceptic – or the dogmatist at a moment of doubt – can ask questions of the following sort: Are the primitive terms really primitive? Are the primitive truths really primitive? Are the rules of inference really safe? “These questions”, Lakatos explains, “played a decisive role in Frege’s and Russell’s great enterprise to go back to still more fundamental first principles, beyond the Peano axioms of arithmetic” (id., p. 11).38 It is this line of thought which backs up

38 As this quotation shows, Lakatos considers only the case of arithmetic and thus ignores the fact that the early Russell’s logicism was not restricted in this way. It would, indeed, be difficult to reconcile Lakatos’ reading of logicism with Russell’s inclusion of geometry within the scope of logicism. Whatever else is said of Russell’s treatment of geometry in the Principles, it is at least clear that it was not his intention to give geometrical knowledge an absolutely certain and unshakeable foundation. It follows from his treatment of geometry that a distinction must be drawn between pure and applied geometries. As regards the former, “all geometrical results follow, by the mere rules of logic, from the definitions of various spaces (1903a, §434). The latter in turn has to do with the problem of which of the abstractly definable geometries correctly describes the properties of physical space. If “geometrical knowledge” is taken to refer to knowledge of applied geometry, it is (partly) empirical and falls outside the scope of logic and logicism. And if it is taken in the former sense, as what follows from which definitions, it is not Ch.1 Preliminary Remarks on Russell’s Early Logicism 51

Lakatos’ further claim that the Euclidean programme implies the “trivialization of [mathematical] knowledge” (id., pp. 4-5); since the logicist can suppress scepticism only by a successful identification of a foundation for which the demand of a further warrant cannot arise, the ultimate foundation can consist only of such truths, principles and terms that must be recognised as “trivialities”. The weakest point in Lakatos’ rational reconstruction is the claim that Russell’s logicism involves a “logico-trivialization of mathematics”. As I read him, this is implied by Lakatos’ view on the “inner dialectic” of the Euclidean programme: nothing short of a foundation with an epistemically trivial load can be considered a viable foundationalist answer to the sceptical argument from an infinite regress. Hence, since logicism is an instance of this programme, the logicist foundation must be construed accordingly. This inference is where Lakatos’ reconstruction invites an immediate objection. Apart from his own conception of what the Euclidean programme requires, there is no reason to think that the early Russell would have attributed anything like vacuous truth or triviality to logic. And there are very good reasons for thinking otherwise.39 That is, Russell’s acceptance of something like the triviality-thesis was not part of his original logicism. In fact, it came only much later and was largely due to Wittgen- stein’s influence.40 It was through him that Russell came to the conviction that logic is in some sense a matter of linguistic rules and hence (again, in some sense) tautologous and “trivial”. Furthermore, Russell says that it was only with great reluctance that he finally accepted the triviality of logical and, therefore, of mathematical truth (1959, pp. 211-12; cf. also Russell 1950-2). This observation suffices to undermine Lakatos’ reconstruction, which presents Russell’s logicism was directed against the specific target of scepticism arising from the classical infinite regress argu- clear that self-evidence has any such role to play in this knowledge as Laka- tos finds for it in the case of arithmetic and its reduction to logic. 39 Cf. here Hager (1994, pp. 40-41). 40 As Russell himself says explicitly (1959, pp. 112-119). 52 Ch.1 Preliminary Remarks on Russell’s Early Logicism ment. There is, however, a more general difficulty besetting any attempt to read Russell’s philosophy of mathematics as a species of epistemic logicism; notwithstanding his later testimonies, issues pertaining to epistemology simply do not surface in the relevant writings (most notably of course in the Principles of Mathematics).41 Russell’s writings from the early logicist period have very little to say about the epistemology of mathematics. If his reasons for occupying himself with the foundations of mathematics had been epistemological in some relevant sense, then, presumably, he would have addressed various questions about mathematical knowledge. But he does not do so. Instead, he gives the following explanation of the objects of his own work (this is from the Preface to the Principles). Firstly, he attempts to establish the thesis that “all pure mathematics deals exclusively with concepts definable in terms of a very small number of fundamental logical concepts, and that all its propositions are deducible from a very small number of fundamental logical principles” (p. xv). The various parts of this thesis are defended “against such adverse theories as appeared most widely held or most difficult to disprove” (ibid.), a task that involves the presentation of “the more important stages in the deductions by which the thesis is established” (ibid.) Secondly, he discusses the fundamental logical concepts and principles which underlie the purported deduction. In his often quoted words, this “discussion of the indefinables [...] is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of a pineapple” (ibid.) This description of the aims of the book makes no mention of anything that has to do with our knowledge.42 The only apparent

41 This means of course that Russell’s later descriptions of his own logicism tend to obscure its real import. 42 It is also worth contrasting two descriptions that Russell himself gave of the development which lead him to write the Principles. In My Philosophical Development he tells that “as a boy” his interest in mathematics was fuelled by practical goals and ambitions (1959, p. 208). Later, however, this gave way to an interest “in the principles upon which mathematics is based” (id., p. 209). Ch.1 Preliminary Remarks on Russell’s Early Logicism 53 exception is the somewhat casual reference to acquaintance that occurs in one of the passages quoted above. Exactly how this term should be understood in this particular context, however, is a less than straightforward matter. It could be taken as an indication of the epistemology which could with some plausibility be read into the early Russell and which surfaces in his later writings; this epistemology builds, in one way or another, on the immediate cognitive relation of acquaintance.43 From this reading there is but a short step to epistemic logicism. That is to say, this interpretation of Russell’s position, if spelled out, would be likely to yield the familiar foundationalist picture of justification, combined with platonism about mathematical objects. The latter portraits mathematical truths as

According to his testimony, the change arose from a wish to combat “mathematical scepticism” (ibid.) Its possibility was suggested to him by the obviously fallacious proofs which his mathematics teachers at Cambridge wanted him to accept and which could be overcome only by finding “a firmer foundation for mathematical belief” (ibid.) It was this kind of research, he writes, which eventually lead him to mathematical logic. This recollection should be compared to how the origin of the Principles was described in that very work: “About six years ago [sc. around 1896], I began an investigation into the philosophy of Dynamics. I was met by the difficulty that, when a particle is subject to several forces, no one of the component accelerations actually occurs, but only the resultant acceleration, of which they are not parts; this fact rendered illusory such causation of particulars by particulars as is affirmed, at first sight, by the law of gravitation. It appeared also that the difficulty in regard to absolute motion is insoluble on a relational theory of space. From these two questions I was led to a re- examination of the Principles of Geometry, thence to the continuity and infinity, and thence, with a view to discovering the meaning of the word any, to Symbolic Logic” (1903a, p. xvi-xvii). Coffa’s distinction between the two senses of rigour is a very apt way to bring out the contrast between these two accounts; Russell, it seems, had largely forgotten about the semantic sense of rigour by the time the series of autobiographical recollections began to flow from his pen. 43 The term “Platonic Atomism” is used by Hylton (1990a) to refer to that set of doctrines which Russell and Moore developed after their rejection of idealism. 54 Ch.1 Preliminary Remarks on Russell’s Early Logicism being about mind-independent, abstract objects. The former exhibits mathematical knowledge as possessing a foundational structure, which divides mathematical truths into two epistemological kinds: those deriving their justification from other mathematical truths, and those the justification of which is immediate or non-inferential (on this view, there is an intimate connection between an axiomatic organisation of a branch of mathematics and our knowledge of that particular branch, and the axiomatisation itself receives its motivation from epistemological considerations, in accordance with epistemic logicism). It is in connection with the second epistemological kind that acquaintance enters the picture; its role is to deliver a crucial part of the foundationalist picture of knowledge, to wit, to render a chain of truths genuinely justificatory by supplying non-arbitrary end-points for the chain. To back up its claim to that role, it is pointed out, firstly, that acquaintance is a direct cognitive relation between the knowing subject and the relevant subject-matter. In this sense it is analogous to ordinary sense-perception; if the justificatory chain is regarded as consisting of beliefs appropriately related to one another, the role of acquaintance can be said to be that of explaining how the justification of mathematical beliefs can terminate with something non-belief-like (mathematical objects and (primitive) truths about these objects). Secondly, to make them suitable for serving as endpoints of justification, acquaintance-based beliefs must be invested with some further property which makes them self-justifying, or justified independently of their relation to other truths. As we have already seen, the standard move here is to resort to self-evidence, which is a sort of propositional correlate of the notion of acquaintance. Since Russell does very little in the Principles to elaborate on the putative epistemological reading of acquaintance, most of the details would remain to be filled in. The use of the word “acquaintance” naturally suggests ideas that are familiar enough; in particular, the justification of mathematical knowledge, according to this view, is in the last instance a matter of a quasi-perceptual connection between Ch.1 Preliminary Remarks on Russell’s Early Logicism 55 the knowing subject and certain objects.44 The issues surrounding this notion need not detain us, since the present point is just to indicate one way to understand Russell’s reference to acquaintance in the passage from the Principles. The suggestion, then, is that this notion has a definite epistemological role to play, one that at least roughly accords with what is suggested by epistemic logicism. I have two objections to the interpretation which presents the early Russell’s philosophy of mathematics as a species of epistemic logicism. Firstly, the epistemic reading of “acquaintance” is not forced upon us. Even though it may be tempting to associate Rus- sell’s talk of acquaintance with epistemology and, in particular, a foundationalist account of mathematical knowledge, other statements that he makes in the Principles about the status of the logical foundations undermine this association.45 To begin with, consider what he writes immediately after the passage about acquaintance:

Where, as in the present case, the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them; there is a

44 Michael Resnik (1980, p. 162) gives the name “epistemic platonism” to this doctrine, according to which “our knowledge of mathematical objects is at least in part based upon direct acquaintance with them, which is analogous to our perception of physical objects”. 45 It is worth pointing out that only a few years after the Principles Russell explicitly rejected the idea or requirement that the fundamental logical premises should possess self-evidence or intrinsic obviousness. This point is made briefly in (1906a, pp. 193-4) and elaborated in his (1907b), where it is argued that the reasons for accepting an axiom are largely “inductive” in character. That is, the reasons for accepting a primitive proposition is not (or is not simply) its self-evidence: “[t]he primitive propositions with which the deductions of logistic begin should, if possible, be self-evident to intuition; but that is not indispensable, nor is it, in any case, the whole reason for their acceptance. This reason is inductive, namely that, among their known consequences (including themselves), many appear to intuition to be true, none appear to intuition to be false, and those that appear to intuition to be true are not, so far as can be seen, deducible from any system of indemonstrable propositions inconsistent with the system in question” (id., p. 194). 56 Ch.1 Preliminary Remarks on Russell’s Early Logicism

process analogous to that which resulted in the discovery of Neptune, with the difference that the final stage î the search with a mental telescope for the entity which has been inferred î is often the most difficult part of the undertaking (1903a, p. xv).

What this passage suggests is that knowledge of the indefinables is not simply a matter of immediate perception, or perception-like process, through which the relevant entities force themselves upon us. The indefinables should rather be described as “inferred entities” in something like the sense that is familiar from Russell’s later writings: they are hypothetical entities postulated for the purposes of fulfilling a certain role. If this is on the right track, it might well be that acquaintance, as Russell uses it in the passage under discussion, has no epistemological connotations at all; it may be that all he is saying is that the early parts of the book are meant to make readers sufficiently familiar with the necessary logical apparatus in order for them to be able to understand what is really going on in the logicist reduction of which an outline is given in the rest of the book. Assuming this is correct, Russell’s use of acquaintance could be compared to what Frege used to call elucidations (Erläuterungen).46 According to Frege, an axiomatic theory must always contain primitive terms, the meaning of which must be understood independently of their role within the theory and which, for that reason, can only be conveyed by means of something that lies outside the theory itself. The role of elucidations is thus the purely pragmatic one of enabling a newcomer to enter the axiomatic system; any means which serve the purpose of creating mathematical understanding and maintaining it can count as an elucidation of a primitive term. There are remarks by Russell that bear a close resemblance to what Frege has to say about elucidations. For one example, consider the following passage with which he opens the manuscript An Analy- sis of Mathematical Reasoning, which he composed in 1898:

46 For Frege’s explanations of elucidations, see, e.g. his (1906a, p. 288) and (1914, p. 224). Ch.1 Preliminary Remarks on Russell’s Early Logicism 57

It is the purpose of the present work to discover those conceptions, and those judgments, which are necessarily presupposed in pure mathematics. It is the habit of mathematicians to begin with definitions î to which axioms are sometimes added î and to assume that definitions, in so far as they are relevant, are always possible. It is, however, sufficiently evident that some conceptions, at least must be, indefinable. For a conception can only be defined in terms of other conceptions, and this process, if it is not to be a vicious circle, must end somewhere. In order that it may be possible to use a conception thus left undefined, the conception must carry an unanalyzable and intuitively apprehended meaning. Intui- tive apprehension is necessary to the student, since he is otherwise unable to understand what is meant; but it is not a logical requisite, being indeed logically irrelevant. All that can be said is, that whoever is desti- tute of this apprehension cannot successfully study the subject in hand, and that any attempt to give him, by means of formal definitions, the conceptions in which he is lacking, is a fundamental error in Logic. Whenever, then, in the present work, such conceptions are discovered, it must not be expected that they should be defined or deduced, but that, as though they were data of sense, they should be merely indicated (1898a, p. 163).47

Since it is not possible to define every term (in the sense of stating the meaning of a term by means of other terms), some terms must be left undefined. But since, as Russell says elsewhere, “[i]t will be admitted that a term cannot be usefully employed unless it means something” (1899a, p. 410), the meaning of a primitive term can only be indicated in a way that suffices for intuitive apprehension. These indications, however, are logically irrelevant in that they do not belong to the axiom-system.48 If the characterisation of a primitive term (its elucidation, as we could call it) serves its purpose, it makes one, as it were, acquainted with the relevant notion or notions which

47 See also Russell’s remarks on definitions in his reply to Poincaré (Russell 1899a, pp. 409-12). 48 Similar-sounding remarks are found in Russell’s description of the Moorean doctrine of the indefinability of Good (Russell 1910, pp. 16-8). There he writes that to answer the question “What is good?” no definition can be provided; what can be given is a characterization which “shall call up the appropriate idea to the mind of the questioner” (id., p. 16). 58 Ch.1 Preliminary Remarks on Russell’s Early Logicism the theory is about. My suggestion, then, is that Russell’s talk of acquaintance, in the Principles, need not be understood as a part of an epistemology of mathematics (along the lines suggested by epistemic logicism). Acquaintance or intuition or intuitive apprehension, it may be argued, is not there to secure the certainty of mathematical knowledge; its role is rather that of giving a somewhat figurative expression to the idea that the logicist reduction cannot be understood unless one has a sufficient grasp of the entities (the so-called logical constants) that, he argues, are to be found in the reductive base. Russell’s recourse to acquaintance is methodological, rather than epistemological. Every theory has its primitive terms and primitive truths. To label them “objects of acquaintance” is merely to indicate their special logical role. Primitive truths, Russell says, “form the starting point for any mathematical reasoning” (1903a, §124). No reason can therefore be given for their truth in the sense of deriving them from other propositions, and in that sense they are objects of immediate apprehension, rather than endpoints of a deduction. What can be done with the primitives is bring them before the mind so that their content and foundational status can be grasped.49

49 This much can, I think, be gleaned from section 124 of the Principles. This section is worth quoting at some length: “The distinction of philosophy and mathematics is broadly one of point of view: mathematics is constructive and deductive, philosophy is critical, and in a certain impersonal sense controversial. Wherever we have deductive reasoning, we have mathematics; but the principles of deduction, the recognition of indefinable entities, and the distinguishing between such entities, are the business of philosophy. Philosophy is, in fact, mainly a question of insight and perception. [—] Such entities [sc. entities which do not exist in space and time], if we are to know anything about them, must be also in some sense perceived, and must be distinguished one from another; their relations also must be in part immediately apprehended. A certain body of indefinable entities and indemonstrable propositions must form the starting-point for any mathematical reasoning; and it is this starting-point that concerns the philosopher. When the philosopher’s work has been perfectly accomplished, its results can be wholly embodied in premisses from which deduction may proceed. Now it follows from the very nature of such inquiries that results may be disproved, Ch.1 Preliminary Remarks on Russell’s Early Logicism 59

This methodological view is of course compatible with a foundationalist conception of mathematical knowledge. That conception, though, does not play a discernible role in the early Russell: whether or not we read an acquaintance-based epistemology into his conception of mathematics, the point of Russellian logicism is not best brought out by considering its possible implications for the epistemology of mathematics. My second objection to the attribution of epistemic logicism to Russell is that it misrepresents how he perceived the relation between philosophy and mathematics. There are, of course, different reasons why someone should be concerned with the epistemology of mathematics. As for Russell, the standard assumption – clearly present in what Lakatos had to say about the subject – is that he needed a proper epistemology to counter scepticism. Rather than examining what Russell did not think, however, I shall turn to an examination of his positive views. When we see what is really involved in rigour, it will also become evident that epistemic logicism misrepresents the meaning that this term had to Russell.

1.3.4 What is Really Involved in Rigour

The point behind Russell’s early logicism is best understood in the light of Coffa’s semantic sense of rigour, rather than the epistemological programme of providing mathematical knowledge with an epistemically unshakeable foundation. This comes out very forcefully in the early sections of the Principles. In the Preface to that work Russell makes it clear that what he is after is a non-arbitrary answer to the question regarding the nature of pure mathematics. In the course but can never be proved. The disproof will consist in pointing out contradictions and inconsistencies; but the absence of these can never amount to a proof. All depends, in the end, upon immediate perception; and philosophical argument, strictly speaking, consists mainly of an endeavour to cause the reader to perceive what has been perceived by the author. The argument, in short, is not of the nature of proof, but of exhortation”. 60 Ch.1 Preliminary Remarks on Russell’s Early Logicism of answering this question, he also comes up with answers to a whole host of other similar-sounding problems: the nature of number, of infinity, of space, time and motion, and of mathematical inference (1903, §2). These are all problems that are semantic in Coffa’s sense: in each case Russell professes to give a “clear account of the basic notions of a discipline” (Coffa 1991, p. 27). Furthermore, these questions or problems are alike in an important respect. In earlier times they had been subject to endless philosophical debate and the uncertainty that is typical of such controversies. As Russell sees it, this unsatisfactory state of affairs had been brought to an end through mathematicians’ efforts. The problems addressed in the Principles are no longer specifically philosophical; instead, they are transformed into problems that can be given a mathematical treatment. This change has important consequences. Minimally, it follows that philosophers, unless they are willing to take the risk of being rele- gated to the category of the scientifically ignorant, must take into account the definitive results of mathematical research in treating any topic on which mathematics can be brought to bear. As Russell puts it elsewhere:

In the whole philosophy of mathematics, which used to be at least as full of doubt as any other part of philosophy, order and certainty have replaced the confusion and hesitation which formerly reigned. Philoso- phers, of course, have not yet discovered this fact, and continue to write on such subjects in the old way. But mathematicians [—] have now the power of treating the principles of mathematics in an exact and masterly manner, by means of which the certainty of mathematics extends also to mathematical philosophy. Hence many of the topics which used to be placed among the great mysteries î for example, the natures of infinity, of continuity, of space, time and motion î are now no longer in any degree open to doubt or discussion. Those who wish to know the nature of these things need only read the works of such men as Peano or Georg Cantor; they will there find exact and indubitable expositions of all these quondam mysteries (1901a, p. 369)

The contrast to which Russell alludes in this passage – one between endless and apparently fruitless philosophising and progressive Ch.1 Preliminary Remarks on Russell’s Early Logicism 61 mathematical research capable of yielding definitive results – could not be made any clearer. It shows, above all, that the import of his logicism should not be seen in the lights of epistemic logicism. As he sees it, mathematics does not need philosophy to safeguard its truth or certainty, which would be in jeopardy in the absence of support from philosophy. On the contrary, it is mathematics which should show the way for philosophers. It is a distinctive gain if a topic can be given a mathematical treatment, since that resolves at once the uncertainty and doubt characteristic of philosophy: “[f]or the philosophers there is [...] nothing left but graceful acknowledgment” (1901a, p. 367), once mathematicians have taken over a topic and shown how to treat it with all the precision and exactitude that mathematics is capable of. This attitude has two consequences, one of them constructive, the other critical. The first has to do with the status Russell assigns to logicism. As he sees it, logicism is not something extraneous to actual or real-life mathematics. In asserting his own version of the logicist thesis, he does not regard himself as advancing a philosophical and hence at least potentially controversial thesis about of the nature of mathematics; on the contrary, logicism is presented as an integral part of the “rigorous mathematics” which was being developed by such mathematicians as Dedekind, Weierstrass, Cantor and Peano. Logi- cism, that is to say, stands closer to mathematics than it does to traditional philosophical accounts of the nature of mathematics. That this is Russell’s attitude is borne out by what he writes in section 3 of the Principles. There he gives a brief description of the benefits which accrue to the philosophy of mathematics from the logicist reduction:

The philosophy of mathematics has been hitherto as controversial, obscure and unprogressive as the other branches of philosophy. Although it was generally agreed that mathematics is in some sense true, philosophers disputed as to what mathematical propositions really meant: although something was true, no two people were agreed as to what it was that was true, and if something was known, no one knew what it was that was known. So long, however, as this was doubtful, it could hardly be said that any certain and exact knowledge was to be obtained in 62 Ch.1 Preliminary Remarks on Russell’s Early Logicism

mathematics. [...] This state of things, it must be confessed, was thoroughly unsatisfactory. Philosophy asks Mathematics: What does it mean? Mathematics in the past was unable to answer; and Philosophy answered by introducing the totally irrelevant notion of mind. But now mathematics is able to answer, so far at least as to reduce the whole of its propositions to certain fundamental notions of logic. At this point the discussion must be resumed by Philosophy. I shall endeavour to indicate what are the fundamental notions involved, to prove at length that no others occur in mathematics, and to point out briefly the philosophical difficulties involved in the analysis of these notions. A complete treatment of these difficulties would involve a treatise on logic, which will not be found on the following pages.

The general tone of this passage is already familiar. The point to be emphasised now is that Russell presents logicism as the answer that mathematics gives to philosophical queries about the nature of mathematics (and related subjects). The importance of logicism, that is to say, lies in the fact that it shows how the hitherto obscure subject known as philosophy of mathematics can be replaced by a new discipline î one that could be called “mathematical philosophy” î which differs in no way from mathematics as far as its exactness or finality is concerned.50 What logicism effects is the transformation of philosophical problems into problems of logic. As Russell also indicates in the above quotation, the “fundamental notions” to which the problems of the philosophy of mathematics are reduced present fresh difficulties of their own, so that the outline of logicism presented in the Principles is characterised by a certain incompleteness; in addition to the obvious incompleteness which results from the fact that the derivation of “pure mathematics” from logic is not given in detail, there remains the additional task of presenting a detailed account of

50 What justifies the application of the word “philosophy” to logicism is the fact that logicism deals with the foundations of mathematics, or those principles which underlie mathematical sciences. These are topics which were formerly left to philosophers. There is thus a “topical continuation” between traditional philosophical theories of the nature of mathematics (like the one that Kant developed) and Russell’s logicism. Ch.1 Preliminary Remarks on Russell’s Early Logicism 63 logic itself, a task that Russell shuns in the book. This latter incompleteness, however, does not undermine the significance that he attaches to logicism.

1.4 Conclusions: Russell and Kant

Russell’s conception of the status of mathematics-cum-logicism vis-à- vis (traditional) philosophy has an important critical function as well, as it makes the former a very effective means of philosophical criticism. Suppose that some philosophical theory turns out to conflict with some part of established mathematics (and, Russell argued time and again, what idealists had to say about mathematics and related subjects offered any number of illustrations of this clash). This could happen in either of the following two ways: either the theory is in straightforward contradiction with an established mathematical result or else it cannot make room for some mathematical phenomenon.51 Either way, Russell’s conclusion would be that it is the philosophers’ views which must be rejected or at least modified so as to bring them into harmony with acknowledged mathematical facts. Furthermore, those philosophies which he knew were, by and large, indifferent to what had taken place in mathematics itself. This was either because they were tied up with pre-nineteenth century mathematics or else because they simply ignored what mathematicians had discovered

51 A an example of the former case is the idealist Russell’s rejection of Cantor’s transfinite arithmetic (see Russell 1896a, p. 37; 1896b, pp. 50-2, 1897a; cf. Griffin 1991, pp. 240-3; G. H. Moore 1995, Levine 1998, pp. 96- 8). If, as Russell did at the time, numbers are looked on as something that results from counting or synthesis, it follows immediately that a “completed” or actual infinity stands condemned, since, as he explains, “absolute infinity is merely the negation of possible synthesis, and thus the negation of number” (1896b, p. 50). An example of the latter case is Kant’s theory of geometry, which, if unmodified, comes to conflict the mathematical fact that there are several internally consistent but mutually inconsistent geometries (this claim, though in my opinion correct, is of course controversial; for a forceful defence of it, see Friedman (1992, ch. 2). 64 Ch.1 Preliminary Remarks on Russell’s Early Logicism about their own discipline.52 From his newly acquired perspective, all these philosophies had been rendered obsolete by progress in mathematics. For Russell, Kant’s theory of mathematics occupied a special place among these philosophies. His own gradual progress towards the logicist position could be described roughly as a transition, both mathematically and philosophically, from an essentially pre- nineteenth century understanding of mathematics to a full appreciation of mathematical rigour. Initially, his knowledge of mathematics was based on what he had learned as a Cambridge undergraduate in mathematics, and his earliest attempts within the philosophy of mathematics were conceived within a broadly idealistic framework which owed a great deal to Kant.53 In the course of his subsequent

52 Thus, for instance, mathematics was no longer describable as the science of “discrete and continuous quantity”, which had been the standard characterisation up until the early nineteenth century (and remained so at least for many philosophers for decades to come (as Russell (1901a, p. 376- 7) in fact points out). As Bolzano wrote in 1810: “in all modern textbooks of mathematics the definition is put forward: mathematics is the science of quantities (1810, §1). Bolzano, of course, went on to criticise variants of this definition for being either too wide (sloppy or off-hand definitions of “quantity” did not give a distinguishing characteristic of mathematics) or else too narrow (the quantity view excluded important regions of mathematical research). 53 Griffin (1991) is the first detailed study of Russell’s idealism. It contains, among other things, an elaborate description of Russell’s mathematical development in the 1890s. For the evolution of Russell early logicism, see Rodríguez-Consuegra (1991). Though brief, Levine (1998) is very informative and illuminating on transitional period in Russell’ development from idealism, through Moorean logic and metaphysics, to Peano and the Principles of Mathematics. Kant’s influence on Russell is best seen in Russell’s Essay on the Foundations of Geometry of 1897, the substance of which consists in an endeavour to bring a broadly Kantian position in line with the existence of non-Euclidean geometries. In 1898, while working on a manuscript entitled An Analysis on Mathematical Reasoning, he could still write to Louis Couturat: “I am asking the question from the Prolegomena, ‘Wie ist reine Mathematik möglich?” I am preparing a work of which this question could be the title, Ch.1 Preliminary Remarks on Russell’s Early Logicism 65 development Russell absorbed more and more of the work which was being done on the continent and which he associated with such names as Weierstrass, Dedekind, Cantor and Peano î all of them advocates of “rigorous mathematics”. With his growing realisation that the mathematics which was being done on the Continent was very far from the mathematics which he had been taught at Cam- bridge, he also came to see that he had been looking in the wrong direction for a philosophical account of mathematics. These developments in Russell’s view have been very well summed up by Nicho- las Griffin: “[o]ne of the things which makes Russell’s development in the 1890s so interesting is that, within the space of seven years, he moves from a full-blooded Kantian position, such as might have been widely accepted at the beginning of the century, to a complete rejection of Kant, a position which was not common even among the advanced mathematicians of the time”.54 The facts about Russell’s own development that Griffin mentions in this passage, to wit, that Russell’s own efforts in the philosophy of mathematics had been inspired by Kant, and that he gradually fought himself free from all even remotely Kantian elements in his own philosophy, make it easy to understand why he was, once he had formulated his logicist position, so eager to spell out the consequences of “modern mathematics” for Kant’s philosophy. The logicist Russell’s interest in Kant’s theory of mathematics is thus a natural outcome of his own development. Kant’s theory, he came to think, was tied up with an outmoded conception of mathematics in exactly the same way that his own efforts had been. Under- stood in this way, Kant’s position is vulnerable to an obvious and immediate charge. Russell was not slow to make use of it (and his own development certainly made him very sensitive to it); if Kant’s theory of mathematics reflects the mathematics of his day, it is unlikely to stand unscathed if mathematics itself changes. And cer- and in which the results will, I think, be for the most part purely Kantian” (letter to Louis Couturat, 3 June 1898; quoted in Russell 1990, p. 157). 54 Griffin (1991, p. 99). 66 Ch.1 Preliminary Remarks on Russell’s Early Logicism tainly it was not far-fetched for Russell, or for anyone else, to think that the whole science of mathematics had been revolutionised in the course of the nineteenth century. If, then, as Russell said, “pure mathematics” was only discovered in the nineteenth century (1901a, p. 366), all earlier attempts – Kant’s theory included – were marred by sheer ignorance as to the subject with which they thought they were dealing with:

This science [sc. pure mathematics], like most others, was baptised long before it was born; and thus we find writers before the nineteenth century alluding to what they called pure mathematics. But if they had been asked what thus subject was, they would only have been able to say that it consisted of Arithmetic, Algebra, Geometry, and so on. As to what these studies had in common, and as to what distinguished them from applied mathematics, our ancestors were completely in the dark. (ibid.)

Russell’s criticism of Kant is a straightforward application of this charge. Russell’s attitude can be illustrated briefly and in a preliminary fashion by the special case of geometry. On the face of it, traditional or pre-nineteenth century formulations of Euclidean geometry were radically different from the geometry of the late nineteenth century. The “second birth” of mathematics put Russell in a position to argue that the distinctive features of the traditional geometrical practice, in particular its notorious “reliance on figures” in the proofs of theorems, were simply due to the fact that geometry had not been developed with full rigour. This in turn had immediate consequences for Kant’s theory of geometry. The latter was centrally concerned with providing an explanation of the possibility of geometry. Given the inadequacies of the mathematics of his day, of which the peculiarities of Euclid’s axiomatisation provided one instance, Kant was bound to be led astray in his purported explanation. That is, Russell thought that it could be demonstrated, thanks to rigorous mathematicians, that Kant’s diagnosis of the conceptual situation was simply an error due to ignorance. This diagnosis, furthermore, could be easily disposed of by looking at the most advanced and carefully formulated Ch.1 Preliminary Remarks on Russell’s Early Logicism 67 mathematical theories of the time. Thus, Russell wrote about Peano and his disciples: “I liked the way in which they developed geometry without the use of figures, thus displaying the needlessness of Kant’s Anschauung (1959, p. 72). In general, Russell argued that changes in mathematics called for a new philosophy as well; this new philosophy, furthermore, would be radically incompatible with anything that Kant had ever said or taught. Russell’s criticism of Kant’s theory of mathematics naturally focuses on the notion of Anschauung or intuition. As the secondary literature on Kant amply demonstrates, there are considerable difficulties in saying what this notion amounts to.55 Whatever one’s preferred answer is to this matter, the important point now is that in arguing for his positive theory, Kant was able to draw on several mathematical practices of his time. From a modern point of view, or from that perspective which Russell came to know in the late 1890s, one may feel the temptation, as Coffa has put it, “to compare Kant’s discovery of pure intuition [...] with Leverrier’s discovery of Vulcan” (1981, p. 33).56 Nevertheless, as Coffa goes on to point out, this “comparison is unfair to Kant in that there were mathematical data that could well be interpreted as justifying Kant’s conclusions” (ibid.) Examples – and these play a major role in Kant’s own expositions of his theory – are provided by the Newtonian understanding of the calculus, the proof-structure of Euclid’s geometry, and accounts of arithmetic which attempt, in one way or another, to build that science on calculation. From the standpoint of the late 19th century, all these data belongs to a superseded stage of mathematical research. Hence, Russell argues, Kant’s theory of mathematics with its notion of pure intuition has become equally outdated; with the data gone, pure intuition (i.e., Kant’s explanation for the data) is also dispensed with. Russell’s case against Kant thus rests on the charge that the latter’s theory of mathematics had been grounded in a mathematical practice

55 This issue is dealt with, e.g., in several of the essays collected in Posy (1992). 56 That is, a discovery of what is not there in the first place. 68 Ch.1 Preliminary Remarks on Russell’s Early Logicism which was insufficiently explicit about its own foundations; this practice, that is to say, had neglected the task of analysing its concepts, judgments and reasonings. When nineteenth century mathematicians carried out this conceptual and methodological revision, thereby producing a completely new picture of mathematics, it began to seem as if Kant’s theory of mathematics had been conditioned by sheer ignorance (“ignorance” in the somewhat peculiar sense that Kant had failed to anticipate the major mathematical and logical discoveries made by later generations). Logicism was not just an exhortation to do mathematics in a rigorous manner; it was also, and above all, a substantial thesis concerning the true nature of mathematics. Kant had tried to make sense of that nature by developing a conception of mathematical method in which the notion of pure intuition played the central role. Further- more, when he explained how his method could yield apriori knowledge, he ended up in transcendental idealism. The logicist Russell subjected these results to a criticism in which the following elements can be discerned. Firstly, his immediate target was Kant’s theory of mathematics and the notion of pure intuition. And the philosophical point of Russell’s logicism lies precisely here; if successful, it shows that and why there is no room for anything like Kantian intuitions in the proper treatment of pure mathematics. Secondly, considerations pertaining to mathematics paved the way for a more general attack on transcendental idealism, or those far-reaching consequences which Kant had drawn from what he thought was needed to understand mathematics and mathematical knowledge. The distinctive features of Kant’s transcendental idealism are connected with the introduction of a new category of judgments, the synthetic apriori, together with his explanation of how there can be such judgments. The underlying theme of Russell’s anti-Kantianism, of which logicism is the cornerstone, is his attempt to undermine Kant’s purported explanation of the synthetic apriori; in those cases where Kant claims we are dealing with synthetic apriori judgments and which, for that reason, involve pure intuition, Russell seeks to show that what is really involved is something purely logical. To draw Ch.1 Preliminary Remarks on Russell’s Early Logicism 69 substantial and non-arbitrary conclusions from this contrast, he had to have a fairly sophisticated conception of logic. This conception can thus be elucidated by scrutinising what Russell has to say about Kant’s theory of mathematics and his transcendental idealism.

Chapter 2 Kant on Formal-logical and Mathematical Cognition

2.0. Introduction

In this chapter, I will develop in some detail a contextualist interpretation of Kant’s theory of mathematics. We have already seen that a claim about the relevant mathematical context plays an important role in Russell’s argument against Kant’s theory: on Russell’s view, Kant’s introduction of pure intuition was entirely dependent upon certain peculiarities of an earlier mathematical practice, which modern mathematics and its pursuit of rigour had rendered obsolete. This chapter can thus be regarded as an elaboration of what we may call the “Russellian” reading of Kant’s theory of mathematics. In developing the contextualist interpretation I will not, however, focus on how Russell understood this dependence (this topic will be taken up in chapter three). Instead, I will give a picture of Kant’s theory and its mathematical background which is more detailed than Russell’s somewhat concise remarks on these topics, but which nevertheless follows the basic underlying idea. This picture will then be put to use in chapter three, in which the specifics of Russell’s criticism of Kant will be considered. As in any interpretation of Kant’s conception of mathematics, contextualist or otherwise, the focus will be on the notion of pure intuition. For the correct appreciation of this notion, it is essential to see that Kant ascribes two distinct functions to it, or uses it to answer two distinct questions. Firstly, there is the familiar transcendental or epistemological question concerning the applicability of mathematics to empirical objects: How can we know or independently of experience that mathematical judgments, which are synthetic, are true of the empirical world? In focusing on this question, Kantian scholars have often neglected the second function that Kant attributes to pure intuition. For he uses it, in effect, to solve a semantic problem, or prob- 72 Ch. 2 Kant on Formal-Logical and Mathematical Cognition lem that concerns content.1 Very briefly, Kant argues that a reflection on how mathematics is done shows the content of mathematical concepts to be inextricably tied up with something he calls “construction in pure intuition”. This semantic problem of explaining the content of mathematical concepts (and judgments and reasoning) is independent of the transcendental or epistemological problem, even though Kant’s resolution of the semantic problem is a necessary ingredient in his eventual answer to the latter problem. The distinction between the two problems is important for two reasons. Firstly, ignoring the distinction makes one apt to misconstrue Kant’s theory of mathematics (plus much else that is found in the first Critique). Secondly, the distinction enables us to give a clear picture of the relation that Kant’s theory bears to subsequent philosophies of mathematics (and here, of course, I have Russell in mind); in particular, it enables us to keep separate the different questions which such philosophies were calculated to answer. I should note at the outset that the interpretation outlined in this chapter is largely derivative. In particular, I draw extensively on Mi- chael Friedman’s interpretation of Kant, both on a general level as well as in questions of detail.2 Friedman’s starting point is that Kant’s

1 The term “semantic” must be construed here in a broad î or, perhaps, pre-Tarskian î sense (this broad sense is familiar from Coffa 1991). In this sense anyone with an interest in “representation”, “content”, etc. is en- gaged in “semantics”. Semantics in this broad sense is not necessarily tied to language (and even less to formal semantics); these associations emerged only in the 1920s and 1930s, when certain philosophers and logicians effected the linguistic turn. Instead of “semantic”, we could use the more neutral term “representation-theoretic”, which Richardson (1998, Ch. 4) uses for a similar purpose. 2 See Friedman (1992a, Ch. 2 and Ch. 3), (2000). Young (1994) may also be mentioned in this connection. Kant’s mathematical background is studied in detail in Shabel (2003), to which I am indebted. It is commonplace that Kant’s theory of geometry is heavily influenced by Euclid’s Elementa (see, e.g. Hintikka 1967, 1969a), Brittan (1978)), but in scrutinising his remarks on arithmetic and algebra, scholars have usually overlooked the fact that there is, in fact, a fairly straightforward connection between Kant and “mathematical tradition” on this point. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 73 distinction between concepts and intuitions is one between what can and cannot, respectively, be represented by means of the logical forms recognised by traditional logic. Since the very formulation of mathematical theories about such topics as continuity and infinity requires, as we might put it, polyadic predicate logic, Kant concluded, using the standards available to him, that their representation requires intuitive rather than purely conceptual resources. This observation, which in one form or another goes back to at least Russell (1903) and Cassirer (1907; 1910), provides the starting-point for the interpretation presented in this chapter as well.

2.1 Kant’s Programme for the Philosophy of Mathematics

2.1.1 Preliminary Remarks

Kant’s remarks on mathematics are primarily intended to serve the negative or critical purpose of showing that there is an unbridgeable gap between mathematical and philosophical method. Clearly, such a claim would have little force unless it was backed up by a fairly detailed conception of the essentials of the method underlying mathematical knowledge. Thus, what Kant says about mathematics provides at least an outline of a philosophical reconstruction of mathematical knowledge. Any reconstruction of this kind must deal with two problems. Firstly, it must deliver an answer to the semantic problem of spelling out the content of the different sorts of statements or judgments that are made about the subject-matter at hand (in this case, some branch of mathematics).3 Secondly, there is the epistemological problem of ex-

3 Here I speak simply of statements (judgments, etc.), but the spelling out of “content” may involve a lot more complicated structures than single statements or judgments. In the case of mathematics, the semantic task is likely to involve the articulation of nothing less than the structure of mathematical theories (in particular, the structure of mathematical proofs), since mathematical knowledge is capable of being organised in the form of a systematic theory. 74 Ch. 2 Kant on Formal-Logical and Mathematical Cognition plaining how we can have knowledge of the subject-matter, once we have fixed upon some particular account of its semantics. The reconstruction of mathematical knowledge is as a rule carried out with a view to salvaging some property which is assumed, independently of the purported reconstruction, to characterise such knowledge. Notwithstanding some dissidents like J. S. Mill, philosophers have traditionally assumed that a theory of mathematical knowledge must abide by the constraint of apriority. Kant’s theory of mathematics provides a clear example of this assumption; as he explains in the Critique of Pure Reason, “[m]athematics presents the most splendid example of the successful extension of pure reason, without the help of experience” (A712/740). His theory is thus, among other things, an attempt to vindicate the apriori character of mathematical knowledge, and he is, accordingly, committed to what Philip Kitcher has dubbed the apriorist program.4 For Kant, this vindication was made particularly urgent by the fact that he thought to have found good reasons to think that there was in earlier models of apriori knowledge a mismatch between the semantic and epistemological components. He held, furthermore, that mathematics provided one crucial instance where this mismatch was clearly visible. His contention was, then, that earlier models of apriori knowledge – in particular that propounded by Leibniz î revealed their inadequacy through their application to mathematics. This claim concerning the semantics/representational content of mathematical judgments had in turn implications for epistemology, which called for a concomitant revision.

2.1.2 The Leibnizian Background

Kant’s semantic (or, better, representation-theoretic) innovations are best illustrated by comparing them with Leibniz’s proto-semantics. This seems to have been the reference-point which Kant had in mind

4 See Philip Kitcher (1983, Ch. 2), 1986. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 75 when he explained his discoveries.5 According to Leibniz’s famous inesse-principle, “an affirmative truth is one whose predicate is in the subject”. According to this view of the semantics of propositions – we could call it the conceptual containment model – every proposition is either explicitly or implicitly identical; confining attention to categorical and affirmative propositions, the contention is that every proposition of the form “A is B” (read: predicate B belongs to subject A) reduces ultimately to one of the form “AB is B”, which shows that the predicate is a part of, or is contained in, the subject. Another way to put this is to say that, according to Leibniz, the truth of a true proposition is grounded in the principle of contradiction. One of his formulations of the principle is that identical statements are true and their contradictions are false.6 When we add to this the principle of sufficient reason (principium rationis), we get the two principles which for Leibniz underlie all knowledge. In its most general form the principle of sufficient reason can be expressed in some such form as “there is nothing without reason” or “nothing happens without reason”, but it can also be formulated in a manner that is less metaphysical and more logical in character. Formulated in this way, the principle has to do with the demonstration of truths and it amounts to the requirement that every truth which is not identical (per se true) is provable from such identities. This formulation gives at the same time Leibniz’s definition of truth.7 The logical version of Leibniz’s principium rationis or the contention that every proposition must have an apriori proof (proof that is independent of experience) is likely to strike a less rationalistically inclined philosopher as far-fetched. Nevertheless, it gains some plausibility (or, at any rate, is less obviously false), if it is restricted to the sphere of mathematics.8

5 As is pointed out by Brittan (1978, p. 46). 6 See Kauppi (1960, Ch. II.4) for references and further discussion. 7 See Kauppi (1960, Ch. II.6). 8 Even though the principium rationis is supposed to apply to all truths, its content differs depending on whether the truths to which it is applied are necessary (like mathematical truths) or contingent. Firstly, since contingent truths are about infinitely complex individual substances, the possibility of 76 Ch. 2 Kant on Formal-Logical and Mathematical Cognition

In the case of mathematics, the demand for a proof amounts to this. Since mathematical truths are among the truths of reason (vérités de raison), they are characterised by what Leibniz called “absolute necessity”. And this means that proofs in mathematics rest on the principle of contradiction. That is to say, every mathematical truth – with the exception of explicitly identical truths, which “cannot be proved and have no need of proof”9 – are provable from explicitly identical truths, i.e., reducible to such truths by the substitution of definitional equivalences. That such a proof is capable of yielding a genuine demonstration (and hence knowledge of the demonstrated proposition) depends on two further claims. Firstly, identical truths are per se true, giving an a priori proof for them is beyond the reach of the finite human intellect: it is only God who grasps, with His infinite intellect, such truths. Thus, Leibniz explains, in Discourse on Metaphysics, §8, that God, who sees the individual essence of Alexander the Great, sees in it “the foundation of and reason for all the predicates which can be truly predicated of him”. There- fore God knows apriori that Alexander defeated Darius and Porus. Similarly, he knows apriori whether Alexander died a natural death or whether he died by poison, truths which finite human beings can know only through history and hence aposteriori. Secondly, it follows from the absolute necessity of mathematical truths that their proofs can be grounded, in the manner Leib- niz explains, in the principle of contradiction. Even if there were an apriori proof for a contingent truth like “Alexander defeated Darius”, it could not be so grounded; such a truth is necessary only in the hypothetical sense that it is a consequence of God’s free decision and hence follows by “moral necessity” from the principle that “God will always do the best” (id., §14). What is less than perfect does not imply a contradiction in the logical sense (“Alexander who did not defeat Darius” is not a contradictory notion in itself î it is only incompatible with what takes place in the best of all possible worlds). The principle, therefore, that is operative in such would-be- proofs would have to be stronger than the principle of contradiction. Essen- tially, it would be a principle to the effect that the definition of, say, Alexan- der, on which the proof operates is a real definition. And what makes it a real definition is in the last instance the fact that the notion codifies God’s choice to create the actual Alexander from all possible Alexanders existing in his mind as different complete individual concepts. As Erik Stenius has put it, “the very definition of Alexander qua a real definition is an act of creation, the creation of Alexander” (1973, p. 332). 9 Monadology, §35. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 77 which makes their truth immediately evident. Secondly, the definitions that are made use of in the proof must be real definitions, and not merely nominal definitions; that is, they must be definitions of ideas that are really possible. In the case of mathematical ideas, however, this possibility is said by Leibniz to be immediately evident.10 Leibniz’s views on this matter can be illustrated with the well- known example from his New Essays.11 There Leibniz (or his alter ego Theophilus) gives a proof of the proposition “2 + 2 = 4”. In accordance with what was said above about proofs in mathematics, it can be described as follows. Leibniz assumes as definitions the following: (1) 2 = 1 + 1; (2) 3 = 2 + 1; (3) 4 = 3 + 1. The substitutivity of iden- ticals is taken as an axiom, and the proof itself can be written as follows:

1. 2 + 2 = 4 2. 2 + 1 + 1 = 4 3. 3 + 1 = 4 4. 4 = 4

Here the first line states the proposition to be proved. Each subsequent line is obtained from the immediately preceding one by substituting definitionally identical terms. The proof terminates with an explicitly identical proposition. This is the general schema that ap-

10 New essays, bk. 4, Ch. ii, §1. Elsewhere in the Essays Leibniz illustrates the difference between real and nominal definitions with an example from geometry. To define parallel straight lines as “lines in the same plane which do not meet even if extended to infinity” is to give a merely nominal definition, since it can still be doubted whether such lines are possible. When it is explained, however, that, given a straight line, a line can be drawn with the property that it is at each point equidistant from it, it is at once seen that the thing is possible and why the two lines never meet (id., 3.iii.18). As Kauppi (1960, pp. 106-8) points out, Leibniz distinguishes among real definitions a species, so-called “causal definitions”, which describe a method through which the thing defined is created. Geometrical definitions are precisely of this kind; as Leibniz says, geometers have recognised the need at least to postulate the possibility of things corresponding to their definitions. 11 New Essays, bk. iv, Ch. vii, §10. 78 Ch. 2 Kant on Formal-Logical and Mathematical Cognition plies, according to Leibniz, to all (non-primitive) necessary truths or truths of reason (in particular, to mathematical truths).12

2.1.3 Kant on Analytic and Synthetic Judgments

Let us now turn to Kant. He draws the distinction between analytic and synthetic judgments in a manner which leaves no room to doubt that he had Leibniz’s semantics in mind.13 One of the points he makes with the help of the distinction is that the kind of semantics which Leibniz had advocated could at most be mobilised to explain the truth and apriori knowability of one particular (and all but uninteresting) subset of judgments, the analytic ones. Kant’s departure from the tradition – from Leibniz in particular – is a consequence of his insight that there is a further class of judgments, whose proper understanding requires major semantic or representation-theoretical and, therefore, epistemological innovations. In arguing for the non- emptiness of this class, Kant used mathematics as one example. Clearly, he took it for granted that a simple arithmetical truth like 7 + 5 = 12 is apriori knowable. And yet he denied what Leibniz and others had asserted, namely that its semantics or content could be under-

12 As regards the arithmetical example, there is the standard complaint that the proof makes use of an assumption, the associativity of addition, which is not made explicit; see, e.g. Frege 1884, §6. I shall not dwell on this, since the point is not to assess the merits or otherwise of Leibniz’s conception of mathematical truth and proof, but simply to outline the background of certain Kantian doctrines. 13 Kant’s first explanation of analyticity in the Critique, according to which a judgment is analytic if “the predicate B belongs to the subject A, as something which is (covertly) contained in this concept A” (A6/B10), involves an unmistakable reference to Leibniz’s inesse-principle. Similarly for the second characterisation, which is in terms of the kind of proof that is appropriate for analytic judgments; since every analytical truth is either explicitly or implicitly of the form “(every) AB is B”, their truth can be seen immediately or else can be established by simple substitutions. As Kant says (A150-151/B190-191), analytical judgments rest on the principle of contradiction, which, as we saw, is what Leibniz says about truths of reason. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 79 stood along the lines suggested by the conceptual containment model (see B15-16). This observation extends to geometry as well. The proposition that the sum of the angles of a triangle is equal to two right angles is something that can be known apriori; but, again, Kant thinks he has shown that this piece of knowledge cannot be gleaned from the relevant concepts in the way Leibniz had suggested (see A716-717/B744-745). Kant’s semantic insight establishes that judgments belonging to the second class cannot be characterised as analytic in the sense that he had given to that notion (and sense that would be relevant to assessing Leibniz’s theory of mathematical knowledge). Assuming, as Kant does, that these judgments are nevertheless apriori, leads to the fundamental problem of the first Critique: What explanation can be given of the fact that there are truths which are apriori knowable, even though they are synthetic in the sense of resisting the model applicable to analytic truths? Underlying Kant’s theory of mathematics, then, there is the representation-theoretical view that, in order to understand such non- trivial bodies of knowledge as mathematics, it is necessary to recognise the existence of a new species of representations, “intuitions”. In the case of synthetic judgments, the connection between the subject- concept and the predicate-concept is not one of conceptual containment: if a judgment is synthetic, the connection of the subject and the predicate is not “thought through identity” (A7/B11), as Leibniz might have said. Instead, “in synthetic judgments I have to advance beyond the given concept, viewing as in relation with the concept something altogether different from what was thought in it. This relation is consequently never a relation of identity or contradiction; and from the judgment, taken in and by itself, the truth or falsity of the relation can never be discovered” (A154-155/B193-194). In other words, a synthetic judgment depends for its truth on there being something else outside the judgment which serves to connect the subject-concept with the predicate-concept, or, to revert to Kant’s idiom, something in which “alone the synthesis of two concepts can be achieved (A155/B194). For example, to prove that an arithmetical formula like “7 + 5 = 12” is true, 80 Ch. 2 Kant on Formal-Logical and Mathematical Cognition

[w]e have to go outside these concepts [sc. the concepts of 7 and 5 and of their sum], and call in the aid of the intuition which corresponds to one of them, our five fingers, for instance, or, as Segner does in his Arithmetic, five points, adding to the concept of 7, unit by unit, the five given in intuition. For starting with the number 7, and for the concept calling in the aid of the fingers of my hand as intuition, I now add one by one to the number 7 the units which I previously took together to form the number 5, and with the aid of that figure [the hand] see the number 12 to come into being. That 5 should be added to 7, I have indeed already thought in the concept of the sum = 7 + 5, but not that this sum is equivalent to the number 12. Arithmetical propositions are therefore always synthetic (B15-16).14

2.1.4 The Reasons behind Kant’s Innovation

2.1.4.1 Concepts and Constructions

The logical discoveries of the late 19th century appear to constitute the antithesis to Kant’s doctrines (as Russell argued on several occasions). There is, nevertheless, one significant point of agreement between Kant and the mathematical logicians: both were sharply critical of Aristotelian logic, albeit in very different ways. Kant has often been criticised for an uncritical acceptance of traditional logic, a view apparently licensed by his famous remark that since the time of Aris- totle, formal logic “had not been able to advance a single step”, and is, therefore, in all likelihood, a “closed and completed body of doctrine” (Bviii). There is some truth in this charge, at least to the extent that Kant did not feel any need to criticise that doctrine from the standpoint of formal logic.15 This allegation, however, misses an important aspect of Kant’s thinking. It fails to appreciate the fact that he was well aware of the almost negligible import of traditional logic

14 For the case of geometry see Kant’s discussion in A716-717/B744- 745. 15 Here I ignore such minor, “technical” criticisms as are found, for example, in “On the False Subtlety of the Four Syllogistic Figures” and elsewhere in Kant’s writings. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 81 in dealing with issues which we (or someone like Frege or Russell) would be ready to recognise as matters of logic (sufficiently broadly construed to cover not only such topics as deductive inference but also such as the analysis of the structure of particular kinds of propositions). This point has not been always appreciated, and hence the accusation that with his transcendental logic Kant imported to philosophy that “curious mixture of metaphysics and epistemology which lead to psychologism and the downplay of logic properly so- called” (Kneale and Kneale 1962, p. 355). It is true that this kind of development did take place in the course of the 19th century. And it is likely, furthermore, that the different de-transcendentalised readings of Kant’s transcendental idealism were an important source for the psychologism which was quite popular in the late the 19th century. Yet, in fairness to Kant, it must also be stressed that the introduction of intuition and suchlike had a perfectly respectable source and motivation; he came to see that the Aristotelian model for concepts could not be applied to mathematics and, therefore, that formal logic, as he knew it, could not be used to throw light on the nature of mathematics (this is how the reasons for drawing the distinction between analytic and synthetic judgments were described above). This negative point about the limited conceptual resources or expressive power of traditional logic forms a preliminary step towards a full account of mathematical knowledge, and the eventual theory purports to show that certain recognised properties of mathematical knowledge could be explained only by assuming that mathematical theories are grounded not in concepts simpliciter, but in a certain kind of activity which Kant calls the construction of concepts and which he explains with the help of the notion of intuition. When he explains these notions and develops a new account of the content of mathematical judgments, which is in stark contrast with the Aristotelian model, he is quite clearly occupied with problems which are not far from those that exercised, say, the logicist Russell. The gist of this interpretation of Kant has been given a concise formulation by Ernst Cassirer in his critical appraisal of the logicist philosophy of mathematics and its implications for Kant’s philosophy: 82 Ch. 2 Kant on Formal-Logical and Mathematical Cognition

The division of understanding and sensibility is, in the way it is introduced in the transcendental aesthetic, in the first instance thoroughly convincing: because here it is a matter only of distinguishing mathematical concepts from the general species-concepts, which are defined by genus and difference, of traditional logic. That mathematical definitions flow from pure intuitions means here only that they are not, like the “discursive” concepts of formal logic, abstracted from a multiplicity of different contents as their common marks, but have, instead, their origin in the fully determinate and unique act of construction. This contrast between mere subsumption and the “synthetic” creation of a content is in itself fully clear; but, as can be seen, it concerns only the delineation of the traditional logical technique, and not the new, positive view on concept-formation that Kant himself grounds in his own “transcendental” logic.16

Cassirer makes two points in this passage. Firstly, there is the by now familiar point that Kant’s introduction of what is known as “transcendental logic” serves in the first place the negative purpose of demonstrating the poverty of traditional logic. Secondly, he explains Kant’s reasons for dissatisfaction; Kant had seen that concept- formation in mathematics does not comply with the traditional model incorporated in Aristotelian logic. It was in order to overcome the

16 Cassirer (1907, pp. 32-3). The original German reads: “Die Trennung von Verstand und Sinnlichkeit ist in der Art, wie sie in der transscendentalen Äesthetik eingeführt wird, zunächst durchaus überzeugend: denn hier handelt es sich nur darum, die mathematischen Begriffe von den allgemeinen Gattungsbegriffen der traditionellen Logik, die durch Genus und Differenz definiert werden, zu unterscheiden. Dass die Definitionen der Mathematik der reinen Anschauung enstammen, das bedeutet hier nur, dass sie nicht, wie die “diskursiven” Begriffe der formalen Logik, aus einer Vielheit verschiedener Inhalte als deren gemeinsames Merkmal abstrahiert sind, sondern in einem völlig bestimmten, einzigartigen Akt der Konstruktion ihren Ursprung haben. Dieser Gegensatz zwischen der blossen Subsumption und der “synthetischen” Erschaffung eines Inhalts ist in sich völlig klar: aber er betrifft, wie man sieht, nur die Abgrenzung gegen die herkömmliche logische Technik, nicht gegen die neue positive Auffassung der Begriffsbildung, die Kant selber in seiner eigenen ‘transscendentalen’ Logik begründet”. The translation of the first sentence is taken over from Richard- son (1998, p. 112), and the rest of the passage is my own translation. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 83 shortcomings of the traditional model, Cassirer continues, that Kant introduced sensibility as an autonomous faculty with intuition as its distinctive type of representation.

2.1.4.2 The Containment Model for Concepts

Cassirer’s points can be elaborated by considering the distinction that Kant drew between conceptual and intuitive representations. He accepted a notion of concept that was, by and large, in accordance with the Aristotelian model. In the passage quoted above, Cassirer mentions two salient features of this model.17 Firstly, there is the explanation of the origin of concepts. On the traditional view, concepts have their origin in the process of “universalising abstraction”.18 That is, a

17 These are explained in greater detail in Cassirer (1910, Ch. 1). 18 I borrow the term “universalising abstraction” from Gambra (1996, p. 285). As Gambra points out, philosophical tradition distinguished two variants of abstraction: firstly, universalising abstraction, or the “operation by which we produce general ideas on the basis of other ideas which are similar to each other” (ibid.); secondly, separating abstraction, by which was meant “the operation of distinguishing or discerning some ideas from others” (ibid.) In the Aristotelian tradition abstraction in both of the above senses was important, because it had a substantial explanatory role. Philosophers appealed to abstraction to render intelligible how entities which cannot exist on their own – universals and mathematical entities in particular – can still be legitimately conceived independently of individual things, which are the primary existents (for some details, see Weinberg 1965, pp. 6-12). This same pattern was exploited by nominalists/empiricists as well for their own purposes. Philosophers in either of these camps (Aristotelians and nominalists/empiricists) could argue that two kinds of abstraction are needed. Uni- versalising abstraction accounts for universals î either in the Aristotelian manner, according to which universals exist only in primary substances, or in the nominalist/empiricist manner, according to which universals exist only as “general names” (for the latter account, see Locke’s Essay Concerning Hu- man Understanding, bk. II, Ch. xi, §2, where it is explained that in abstraction the mind separates particular ideas from the “circumstances of real existence” and thereby converts them into general names; without this possibility, every object distinguished in experience would have to possess a name 84 Ch. 2 Kant on Formal-Logical and Mathematical Cognition concept is supposed to result from the power of the mind to reflect on objects which it encounters in experience and compare them with one another so as to select or abstract from them a feature with respect to which they are similar. Secondly, this account of origin leads to the description of a concept as a general representation; a concept classifies objects by dividing them into similarity-classes, and a concept itself is a mark of such a similarity shared by objects falling under it. This explanation is complemented by the further assumption that the abstractionist process of concept-formation always results in determinate hierarchies of concepts. These hierarchies organise themselves according to the notions of genus, differentia and species.19 On of its own and there would have to be endless names). Separating abstraction in turn is needed to account for the prima facie problematic character of mathematical entities (for Aristotle’s views, see Gambra (op. cit) and Heath (1949)). It is the sort of abstraction that enables us to separate, in our intellect, those aspects of things that cannot exist separately in reality. By elimi- nating more and more of a thing’s perceptible qualities, separating abstraction yields the most ‘abstract’ properties, namely continuous and discrete quantities, which are then available for systematic study in geometry and arithmetic. At the time of Frege and Russell, this latter variety of abstraction was still widely appealed to by philosophers and mathematicians alike and Frege in particular spent some time in disposing the idea (see Frege 1884, §§29-44, 1899; cf. Dummett 1991, pp. 83-85). This variety of abstraction does not figure in Kant’s discussion (although it is clear from his conception of mathematical concepts that he would have rejected separating abstraction as an explanation for mathematical entities); the concepts of traditional logic î the terms of propositions which figure in syllogistic reasoning î are ones to which universalising abstraction was taken to apply, hence the focus of Kant’s criticisms. 19 As Cassirer (1910, Ch. 1) points out, the hierarchical picture of the structure of concepts and the abstractionist explanation of their origin are independent of each other. The process of abstraction, as traditionally described, does not put any constraints on its possible outcomes. These constraints come from a different source, when the rudimentary picture of concept-formation is backed up by substantial metaphysical assumptions. As Cassirer puts it, “[i]n the system of Aristotle [...] the gaps that are left in logic [sc. concept-formation] are filled and made good by the Aristotelian metaphysics” (id. p. 7). That is, “[t]he determination of concepts according to its next highest genus and its specific difference reproduces the process by Ch. 2 Kant on Formal-Logical and Mathematical Cognition 85 this view, concepts form a hierarchy, on the top of which lies the supreme, or most general concept, which applies to all objects of comparison. Transition to a new, more specific concept (species) is effected by adding to the higher concept (genus) a new feature (differentia), which characterises only some of the objects belonging to the original similarity-class; conversely, from a given species-concept a more general one is formed by disregarding some feature, and bringing more objects into a similarity-class. For example, if the concept material body is taken as a genus, a species falling under it is distinguished by adding the concept animate, which serves to divide the class of material bodies into two non-overlapping classes (animate and inanimate material bodies). The former class is further divided by dint of the concepts animal and vegetable; the concept animal has under it the concept human, which is arrived at by adding the differentiating feature of rationality, and so on. There are thus two sides to the structure of concepts, corresponding to whether we move downwards or upwards in the hierarchy: the extension of a concept consists in all those concepts that are lower to it in the hierarchy; the intension or content of a concept includes all those concepts that are contained in it – ascending the hierarchy, we see that the concept human being contains (at least) the concepts rational, animate and material body. From the standpoint of traditional logic, a complex concept is essentially a conjunction of its characteristics or partial concepts (those that together form its extension), and concept-formation is simply an act whereby from a given concept a new, more specific one is formed by adding a differentia. Thus, exhibiting the content of a concept in accordance with its logical form or structure is to exhibit it as a conjunction of its partial concepts.

which the substance successively unfolds itself in the special forms of being” (ibid.) This independence, it may be added, is demonstrated by such examples as that provided by Locke. His doctrine of general ideas was thoroughly abstractionist; at the same time, he was thoroughly sceptical of Aristotelian natural kinds. 86 Ch. 2 Kant on Formal-Logical and Mathematical Cognition

2.1.4.3 A Comparison with Frege

In Grundlagen der Arithmetik, §88, Frege wrote that, of all possible ways of forming concepts (Begriffsbildungen), the one that complies with the Aristotelian model is the least fruitful one. He illustrates the difference between the Aristotelian model and the really fruitful definitions (such as his own definition of number or Weierstrass’ definition of the continuity of function) with the help of a geometrical analogy.20 If we represent concepts or their extensions (in our sense, and not the traditional one) by means of what we would now call Venn-diagrams, i.e., by means of overlapping regions in a plane, then a concept defined by common characteristics is represented by an area common to all regions and enclosed by segments of their boundary lines. In this way the Aristotelian model of conceptual analysis – which, it will be recalled, amounts to representing a concept as a logical product of its partial concepts – always makes use of existing boundaries to de- marcate an area, and “[n]othing essentially new [...] emerges in this process” (ibid.) Frege continues the passage by writing that “the more fruitful type of definition is a matter of drawing boundary lines that were not previously given at all.” What can be inferred from this kind of definition “cannot be inspected in advance; here, we are not simply taking out of the box again what we have just put into it”. There is a substantial similarity between the contrast that Frege draws in these passages and the way Kant saw the difference between the Aristotelian model, or the dissection of concepts, and the possibilities inherent in intuitive representation, or the “construction” of concepts. “What we shall be able to infer from it [sc. from a fruitful definition]”, Frege writes, “cannot be inspected in advance; here, we are not simply taking out of the box again what we have just put into it. The conclusions we draw from it extend our knowledge” (ibid.) For Kant, the epistemic value of analytic judgments is merely “explicative” (A7/B11), since they enable us to make explicit what has “all along been thought” in a concept, whereas a synthetic judgment adds to a concept a determi-

20 The analogy is also given in Frege (1880/1, pp. 37-38). Ch. 2 Kant on Formal-Logical and Mathematical Cognition 87 nation which “has not been in any wise thought in it, and which no analysis could possibly extract from it” (ibid.) Frege also remarks in Grundlagen, §88, that Kant “seems to think of concepts as defined by giving a simple list of characteristics in no special order”. This is correct if limited to representations as they are dealt with in Kant’s “formal logic”. However, if it is intended as a general criticism of Kant, Frege’s remark ignores the role of intuitive representations. Indeed, if we follow Cassirer (and some recent scholars21), we can say that recognising the existence of the gap which Frege points out was the very reason that led Kant to his semantic or representation- theoretic innovations. Kant did accept the traditional notion of concept in its essentials.22 This is signalled by his definition of concept as a representatio per notas communes, a representation through common marks or characteristics (The Jäsche Logic, §1), which has its origin in the “logical actus of comparison, reflection, and abstraction” (id., §§9- 11; 15-16), and it is further reflected in his characterisation of the task of “general logic”, according to which “it is to give an analytical exposition of the form of knowledge [as expressed] in concepts, in judgments, and in inferences, and so to obtain formal rules for all employment of understanding” (A132-133/B171-172). On the other hand, Kant’s divergence from tradition becomes evident, when he introduces intuitions as a new species of representation in addition to concepts.23 This is accompanied by the division of human cognitive faculties into understanding (cognition through concepts) and sensi-

21 See Michael Friedman (1992a, Ch. 1 Ch. 2) and J. Michael Young (1994). 22 For Kant’s appropriation of the traditional picture of concepts, see Wilson (1975, pp. 252-3) and Allison (1983, pp. 92-4). 23 In the Jäsche Logic, §1, Kant defines intuition as a individual representation (representation singularis). At A320/B376-377 he explains that subordi- nated to the genus representation stands an objective perception or cognition (cognition; Erkenntnis), which is divided into intuition and concept: “the former relates immediately to the object and is single”, whereas “the latter refers mediately by means of a feature which several things may have in common”. 88 Ch. 2 Kant on Formal-Logical and Mathematical Cognition bility (cognition grounded in intuition), which operate under distinct sets of rules.24

2.1.4.4 Kant on Philosophical and Mathematical Method

That there is a fundamental difference between these two sorts of cognition is a claim which Kant argues for in the Transcendental Doc- trine of Method. There he draws a distinction between “philosophical” and “mathematical” knowledge: the former is purely discursive knowledge, or “knowledge gained by reason from concepts”; the latter he characterises as “knowledge gained by reason from the construction of concepts”, to which he adds the preliminary explanation of “construction” as the apriori exhibition of an intuition which corresponds the concept (A713/B741). This discussion shows, at the same time, that he was well aware of the differences between purely conceptual representation (in his sense) and the “really fruitful definitions in mathematics”, that is, of the inapplicability of the Aristotelian model of concept-formation to mathematics. At A716/B744 this contrast is applied to the special case of geometrical proof or inference. Since a purely logical proof agrees with a “philosophical proof” in that both are analytical derivations or extrac- tions of what is contained in a complex concept, we can here substitute “logical proof” for Kant’s “philosophical proof”. Thus, one case in which the distinction between the two kinds of knowledge is conspicuous is the contrast between purely logical reasoning and mathematical reasoning.

24 “We [...] distinguish the science of the rules of sensibility in general, that is, aesthetic, from the science of the rules of the understanding in general, that is, logic” (A52/B76). The latter science, i.e. “general” or “formal” logic, is formal in the sense that it “treats of understanding without any regard to difference in the objects to which understanding may be directed” (ibid.) It abstracts from sensibility and intuition – from the object- directedness of human cognition – and occupies itself solely with the rules which govern thought (judgment) qua conceptual unity or concept subordi- nation. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 89

Assume, to begin with, that these two types of reasoning agree on their deductive strength or necessity or certainty (to revert to a more Kantian terminology). Assume, that is, that the degree of validity of a piece of mathematical reasoning is no less than that of an argument, like modus ponens, which is valid and whose validity is generally recognised to be a matter of logic – Kant would here speak about “pure general” logic, whose rules apply to all cognition irrespective of its content;25 modus ponens is clearly sufficiently abstract to qualify as a rule of formal logic in Kant’s sense. In this sense, mathematical and logical reasoning are exactly on the same level. Kant, for one, was in complete agreement with this assumption. At A734-735/B762-763 he writes that the main difference between mathematical and philosophical/logical proofs or discursive proofs does not lie in their different degrees of certainty; both are species of what he calls apodeictic proof. Despite this similarity, Kant nevertheless thinks that reasoning in mathematics is not purely logical in character. To show this, he uses as an example Euclid’s proof of the proposition that the sum of the angles of a triangle equals two right angles (A716-717/B744-745).26 When a philosopher attempts a proof of this proposition, all he has at his disposal is a set of given concepts, those that together form the concept of “a figure enclosed by three straight lines, and possessing three angles”. A purely discursive proof, or proof that is sensitive to logical structure only, consists in the activity of detecting and making explicit the constituents of given complex concepts. All that a philosopher or a logician can do, therefore, is to analyse and clarify the concepts of straight line, angle and the number three: “however long he meditates on this concept [sc. of triangle], he will never produce anything new” (A716/B744). By con-

25 See the previous footnote. In what follows, when I speak about Kant’s views about logic, I mean his views about what he recognised as formal logic, that is roughly, everything that belongs to logic according to the Aris- totelian logical tradition. For further discussion of Kant’s views on logic, see section 5.2. 26 “In any triangle, if one of the sides be produced, the exterior angle is equal to the two interior and opposite angles, and the three interior angles of the triangle are equal to two right angles” (Elements, I.32). 90 Ch. 2 Kant on Formal-Logical and Mathematical Cognition trast, when a mathematician is presented with the problem of finding out the relation that the sum of the angles of a triangle bears to a right angle, he “at once begins by constructing a triangle” (ibid.). He then continues by carrying out a series of further constructions on the original figure and draws the relevant conclusions. In this way, “through a chain of inferences guided throughout by intuition, he arrives at a fully evident and universally valid solution of the problem” (A717/B745).

2.1.4.5 Summary

From the Doctrine of Method we can derive two lessons concerning mathematics, one negative, the other positive. Firstly, there is the observation, backed up by reference to existing mathematical practices, in particular to Euclid’s geometry, that the classical model of concept- formation is inapplicable within the region of mathematics. Secondly, there is the positive proposal, to wit, Kant’s version of what was referred to above as the apriorist program: in order to cast philosophical light on current mathematics, and in particular, to explain the apriori character of the knowledge it delivers, an account must be given of the content of mathematical judgments that hinges on the notion of “constructing concepts”.27 Accordingly, we face the following two questions of interpretation: (i) how does the construction of concepts generate (mathematical) contents which go beyond what can be exhibited purely logically? (ii) what are the implications of Kant’s answer to (i) for mathematical knowledge? The first of these two questions is best approached by considering one of the more pressing issues for any interpreter of Kant’s theory of mathematics. This question has to do with the relation between the Doctrine of Method, on one hand, and Transcendental Aesthetic, on the other. The former explains the syntheticity of mathematics as

27 Here, again, I speak simply of the content of mathematical judgments, even though the full articulation of the semantic component of the apriorist program involves more than this. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 91 a function of “constructibility” (and there are of course more than one way to understand this connection28). The latter argues, roughly, that the syntheticity of mathematics can be understood only by the assumption that space and time are apriori intuitions, a claim that, apparently anyway, makes no mention of mathematics as constructive.29

28 For example, Hintikka (1967, 1969a) argues, drawing on Kant’s definition of intuition as a singular representation, that in Kant’s theory of mathematics construction amounts to the introduction of particulars to in- stantiate general concepts. What makes mathematics synthetic, according to Hintikka’s Kant, is the use of instantiation methods, i.e., the fact that mathematical reasoning typically involves the introduction of “new individuals” (as when we use auxiliary constructions in a geometrical proof). A different suggestion is to say that the syntheticity of mathematics is a function, not of the special character of mathematical reasoning, but of its axioms. Thus, Gordon Brittan writes that the reason why mathematics is synthetic, according to Kant, is that it involves propositions which make existential claims (1978, pp. 56-61). Here the contrast is with the propositions of logic, which are “purely formal” and “empty of content”, according to Kant. It should be noted, however, that the difference between a position like Hintikka’s and Brittan’s is really quite minimal (contrary to what Brittan argues). As regards geometry, the relevant existence assumptions are expressed by Euclid’s first three postulates. For Kant, however, postulates are “practical propositions”, which is to say that they have to do with a certain kind of action (in this case construction), and their import is that the action/construction can be carried out (cf. A234/B287). As regards their role in reasoning, they do not function as premises from which consequences are derived (purely logically or “analytically”, as it were). Rather, a postulate is both a starting-point for proof (as in Euclid’s protasis) and that by means of which the proof (or part of it) is carried out. It follows that, for Kant, there is no distinction between axioms and rules of inference in anything like our sense. Brittan writes: “The synthetic character of the propositions of mathematics is a function of some feature of the propositions themselves, and not of the way in which they come to be established” (1978, pp. 55-56). But in the Euclidean context this contrast is really quite spurious. 29 Cf. Butts (1981, p. 267). It must be said, though, that this apparent discrepancy is likely to disappear once it is realised that the Aesthetic is not intended by Kant to stand on its own, but should be read in conjunction with the Analytic; cf. here Walsh (1975, §6). 92 Ch. 2 Kant on Formal-Logical and Mathematical Cognition

2.2 Constructibility and Transcendental Aesthetic

The answer that I prefer to the above question concerning the relation between constructibility and the Aesthetic (what I shall call the “semantic interpretation” of Kant’s notion of construction) can be introduced by contrasting it with a line of thought that is found in many sophisticated expositions of Kant’s theory of mathematics. Ac- cording to this line of thought, the Aesthetic and its doctrine of space and time as apriori intuitions are related only “externally” to mathematics itself. That is to say, the constructibility of mathematics is one thing, to be understood in one way or another on its own; that mathematics is concerned with space and time as the forms of intuition is a claim that emerges only when Kant sets to himself the task of explaining the objective validity of mathematics, or its applicability to empirical objects or appearances.30

30 Hintikka, for example, explains that Kant’s “transcendental problem of the possibility of mathematics” was that of making sense of how constructions î the introduction of particulars as proxies of general concepts î can yield apriori knowledge, i.e., how we can be justified in making certain existential assumptions in the absence of the relevant objects (Hintikka 1969, sec. 16-18; 1984, sec. 3). Kant’s answer to this, according to Hintikka, makes use of this transcendental method: “[s]ince ‘reason has insight only into that which it produces after a plan of its own’ [Bxiii], the explanation of the universal applicability of knowledge obtained by using instantiation methods, i.e., by anticipating certain properties and relations of particulars, can only lie in the fact that we have ourselves put those properties and relations into objects in the processes through which we come to know individuals (particulars). Then the knowledge gained in this way must reflect the structure of those processes, and is applicable to objects only in so far as they are potential targets of such processes” (1984, p. 347; italics in the original). Accord- ing to Kant, Hintikka continues, for beings like us the relevant process is sense-perception. It follows that the properties and relations that mathematics deals with are something that we put into objects in sense-perception, and they are due to the structure of this faculty, i.e., space and time as the forms of intuition. As Hintikka sees it, then, the Aesthetic serves the role of justifying the application of mathematical method to reality. Brittan comes very close to this view. According to him, Kant gives the existential claims made in mathematics (like the postulates of Euclid’s ge- Ch. 2 Kant on Formal-Logical and Mathematical Cognition 93

There are good reasons to maintain, however, that in Kant’s theory of mathematics the connection between constructibility, on one hand, and space and time, on the other, is deeper than that suggested by the above line of thought. Consider first the following quotation from Butts’s exposition of Kant’s theory of mathematics:

To construct an apriori intuition means simply to produce an individual example (in empirical intuition) according to rules of construction that are given by our conceptual system (in this case mathematics). [...] In the case of mathematical constructions the examples are used as representa- tives of universal concepts whose meaning is given in the mathematical system at hand. (1981, p. 269)

If we put it this way, there remains for Kant as well as his interpreters a compulsory question: What is it for a mathematical system to give a meaning to certain universal concepts? The problem that mathematics presents for Kant is not just the one about the empirical applicability of mathematical constructions (although that problem is there, too). Equally important – indeed, more fundamental – is the semantic or representation-theoretic question of how mathematical concepts and judgments receive their content or meaning in the first place. Appreciating the synthetic character of mathematical judgments forces one to recognise an unbridgeable gap between cognition through concepts and cognition grounded in the construction of concepts. To put the point in another way, it forces one to recognise that the content of mathematical judgments cannot be captured by ometry, which Brittan presents as claims “asserting the existence of mathematical individuals” (1978, p. 57), an epistemological twist through the notion of construction (id., p. 66). What is “really possible” – what could be experienced by us – is co-extensive with what is mathematically constructible: “the existence assumptions in Euclidean geometry are justified because of the structure or form of sensibility that in some sense “conditions” physical objects, the existence of which in general we come to know through perception. The same activity that constructs concepts “constructs” objects, and thereby guarantees a fit between them” (id., p. 83; italics added). In a footnote Brittan refers to Hintikka, thus apparently declaring himself to be in agreement with the latter on the import of Kant’s transcendental method. 94 Ch. 2 Kant on Formal-Logical and Mathematical Cognition dint of the simple logical forms recognised in traditional logic. And it is Kant’s eventual explanation of the meaning- or content- constitutive role of construction – mathematical concepts have content only because they can be constructed in pure intuition, which is why these concepts can represent features of objects that are “necessarily implied in the concepts” (Bxii), though not contained in them – that creates the link between the Doctrine of Method and the Aes- thetic (plus other relevant portions of the Critique): the content- constitutive role is grounded, precisely, in space and time as the forms of intuition.31

2.3 The Semantics of Geometry Explained

2.3.1 Constructions in Geometry

To see in more detail what is involved in the semantic interpretation, we may start by considering the case of geometry, which is generally

31 I have formulated the semantic interpretation in terms of the notion of content. It should be noticed that in the Kantian context “content” and “having content” are naturally associated with “objective validity” (and this is how Kant himself uses the German Inhalt, at least on occasion; see, e.g. A239/B298). If the term is used in the latter way, we should say that, according to Kant, a concept has “content” only insofar as object can be given to it in experience: “all concepts [...], even such as are possible apriori, relate to empirical intuitions, that is to the data for a possible experience” (ibid.). This applies to mathematics as well (A239/B299). It is therefore natural to read Kant’s theory of mathematics in a way which makes it a species of “anti-formalism”. On this view, the primary role of intuition-cum- construction is that of securing the truth-in-the-actual-world of a mathematical theory; for this way of using “content”, see Brittan (1978, pp. 61-67); cf. Friedman (1992a, pp. 98-104). On the semantic interpretation, by contrast, “content” has the more familiar meaning of “representational content”, and in its function it is thus comparable to, say, Frege’s notion of judgeable content (beurtheilbarer Inhalt; Frege 1879) or Russell’s proposition. The relation of the two notions of content in the Kantian context is discussed in Young (1994). Ch. 2 Kant on Formal-Logical and Mathematical Cognition 95 regarded as the least obscure part of Kant’s theory of mathematics. Consider, to begin with, the following quotation from Prolegomena, §10:

Now space and time are the two intuitions on which pure mathematics grounds all its cognitions and judgments that present themselves as at once apodictic and necessary; for mathematics must first exhibit all its concepts in intuition, and pure mathematics exhibit them in pure intuition, i.e. construct them. Without pure intuition (as mathematics cannot proceed analytically, namely by analysis of concepts, but only synthetically) it is impossible for pure mathematics to make a single step, since it is in pure intuition alone that the material for synthetic judgments apriori can be given. Geometry is grounded on the pure intuition of space. Arithmetic forms its concepts of numbers by successive addition of units in time; and pure mechanics especially can only form its concepts of motion by means of the representation of time. But both representations are merely intuitions [...].

This passage may not be entirely unambiguous. Kant’s statement that “it is in pure intuition alone that the material for synthetic judgments apriori can be given” could be taken to refer to the fact that the “material” to which we apply mathematics are always objects in space and time. That the point he has in mind is in fact semantic or representation-theoretic is indicated by what he says about arithmetic and mechanics: their characteristic concepts (number and motion, respectively) are themselves inherently temporal in character, i.e., they are grounded in the intuitive representation of time. That there is an analogous dependence of geometry on space can be seen by considering the following three quotations, the first two of which are taken from the Transcendental Analytic.

We cannot think a line without drawing it in thought, or a circle without describing it. We cannot represent the three dimensions of space save by setting three lines at right angles to one another from the same point. Even time itself we cannot represent, save in so far as we attend, in the drawing of a straight line (which has to serve as the outer figurative representation of time), merely to the act of synthesis of the manifold whereby we successively determine inner sense, and in so doing attend to the succession of this determination in inner sense. Motion, as an act of 96 Ch. 2 Kant on Formal-Logical and Mathematical Cognition

the subject (not as a determination of an object), and therefore the synthesis of the manifold in space, first produces the concept of succession [...]. (B154-155; italics in the original)

I cannot represent to myself a line, however small, without drawing it in thought, that is, generating from a point all its parts one after another. Only in this way can the intuition be obtained. [...] The mathematics of space (geometry) is based upon this successive synthesis of the productive imagination in the generation of figures. (A162-163/B203)

All proofs of the complete congruence of two given figures (when one can be replaced at all points by the other) finally come to this, that they coincide with each other; which is obviously nothing other than a synthetic proposition resting on immediate intuition. [...] That complete space (which is not itself the boundary of another space) has three dimensions, and that space in general cannot have more, is built on the proposition that not more than three lines can intersect at right angles in a point. This proposition cannot be shown from concepts, but rests immediately on intuition [...] That we can require that a line shall be drawn to infinity (ad indefinitum) or a series of changes (e.g. spaces passed through in movement) continued to infinity, presupposes a representation of space and time which, in respect of not being in itself bounded by anything, can only attach to intuition; for it could not be inferred from concepts. (Prolegomena, §12)

The import of these passages can be explained in two steps. Firstly, these passages testify to Kant’s acceptance of Euclidean methodology, i.e., the claim that geometry is grounded in the generation of figures. Secondly, Kant argues that this grounding presupposes an intuitive representation of space and time. I shall elaborate on each of these two points in turn. Anyone who thinks, as Kant does, that geometry is grounded in the generation of figures is clearly thinking of Euclid’s Elements. All the geometrical figures with which Euclid is concerned are constructible by suitably combining and iterating certain initial operations, those given by the “construction postulates”:32 (i) to draw a straight

32 The term “construction postulate” is borrowed from Stenius (1978a, §1). Ch. 2 Kant on Formal-Logical and Mathematical Cognition 97 line from any point to any point; (ii) to produce a finite straight line continuously in a straight line; (iii) to describe a circle with any centre and distance. What Kant does is to subject the Euclidean method to a reinterpretation in which construction-postulates (“generation of figures”) is afforded a semantic or representation-theoretic role; that geometry is so grounded means in the first place that geometrical thinking or cognition has a synthetic “origin” or “genesis” in the act of construction.33 And this reference to origin is to be understood in the following sense: were this mode of representation (“drawing of a line in thought” and “describing a circle [in thought]”) not available to us, we could not as much as think of, and reason about, lines, cir- cles and other geometrical entities in the sense that is required if one wishes to do geometry. Kant’s contention, then, is that the acts licensed by the construction postulates must be recognised as an apriori mode of cognition, for it is through construction that the subject- matter of geometry is given to us. This constitutive role of geometrical postulates is made explicit at A234/B287, where Kant uses Euclid’s third postulate as an example:

Now in mathematics a postulate means the practical proposition which contains nothing save the synthesis through which we first give ourselves an object and generate its concept – for instance, with a given line, to describe a circle on a plane from a given point. Such a proposition cannot be proved, since the procedure which it demands is exactly that through which we first generate the concept of such a figure. (Italics added.)

Kant’s view can also be formulated in the following manner. A properly geometrical representation is not a collection of common marks, the analysis of which would show, e.g., that the content of the concept square is constituted by the partial concepts four-sided, equilateral, and right-angled (this articulation amounts to the nominal definition of the concept square). Instead, the semantics of geometrical concepts is given by rules of construction. These specify how geometrical objects

33 As Cassirer puts it in the quotation given above. 98 Ch. 2 Kant on Formal-Logical and Mathematical Cognition of a certain kind, as opposed to mere concepts, are constituted.34 The application of these rules enables us to attach to geometrical figures properties which cannot be shown to belong to them through conceptual analysis. For example, the geometrical properties of a straight line, like its infinite divisibility and extendability, cannot be represented through the mere concept straight line; they can only be represented intuitively, that is, by the construction of an object falling under the concept and the application to it of a further rule of construction, like the bisection of a straight line (Euclid’s proposition I.10) or Euclid’s second postulate. What makes geometrical construction indispensable is above all the fact that the rules underlying such constructions enable us to enter systematic reasoning about geometrical objects and prove general theorems about them which could not be established from concepts alone.35 For the correct appreciation of Kant’s representation-theoretic views it is necessary to observe this rule-governed character of geometrical construction. At A715/B743 Kant writes that “mathematics can achieve nothing by concepts alone but hastens at once to intuition, in which it considers the concept in concreto” (A715/B743). The geometer, then, always puts his rules to use by instancing them, as Kant famously says, “either by imagination alone, in pure intuition, or in accordance therewith also on paper, in empirical intuition” (A713/B741). In carrying out a proof, the geometer makes use of a diagram, say, a triangular figure plus a few additional lines, as in Euclid’s proof of the angle-sum property, which Kant refers to at A716-717/B744-745. And, in order better to keep track of the proof, the geometer may carry out the relevant constructions on a piece of paper. If, however, what he does is an exercise in geometry – if his intention is to produce a proof of a theorem – he does not derive any conclusion from the diagram itself. For the diagram always contains

34 For a detailed description of this contrast between the constitution of concept vs. constitution of object and its importance in the Kantian context, see Young (1994, secs. 5 and 6). 35 A detailed treatment of this idea is given by Friedman (1992a, Ch. 1 and ch. 2). Ch. 2 Kant on Formal-Logical and Mathematical Cognition 99 both more and less than the concept which it is supposed to represent. It contains less because it is not really a triangle but only an approximation; and it contains more since it is not a “mere triangle” but has additional properties.36 Kant says consequently that such a diagram (an empirical intuition) can serve only as an image of the concept triangle.37 And yet this image somehow manages to “express the concept, without impairing its universality” (A714/B742). Kant explains this “somehow” as follows. Geometrical concepts do not receive their content from images in this sense but from what he calls schemata.38 And these he describes precisely as general rules which guide the construction of geometrical figures.39 Thus, the reason why a geometrical proof, even though it is carried out on a particular diagram (as in Euclid’s ekthesis and kataskeue), nevertheless suffices to prove a general theorem, is that in the single figure “we consider only

36 And should I merely imagine the triangle, it would still be more than a mere triangle, as Berkeley pointed out against Locke (Berkeley 1710, §13). 37 Kant’s reason is precisely the one that Berkeley had given: “No image could ever be adequate to the concept of a triangle in general. It would never attain that universality of the concept which renders it valid of all triangles, whether right-angled, obtuse-angled, or acute-angled; it would always be limited to a part only of this sphere” (A140-141/B180). 38 Neglecting the distinction leads to an easy refutation of Kant’s theory of geometry. For example, Louis Couturat (1904, pp. 358-60), referring to Kant’s view that the only way to represent a straight line is to draw it in thought, concludes that his whole theory of geometry consists in confusing geometrical concepts with subjective, mental pictures. To this he then adds the standard critical move which is familiar from anti-psychologistic stric- tures, to wit, that the geometrical properties of straight line are as little dependent on the temporal process of conjuring up mental pictures as they are on the chemical structure of the ink or chalk which I use to draw a straight line on a piece of paper or on blackboard. 39 “Indeed it is schemata, not images of objects, which underlie our pure sensible concepts [sc. mathematical concepts]. The schema of the triangle can exist nowhere but in thought. It is a rule of synthesis of the imagination, in respect to pure figures in space” (A140-141/B180). Immediately before this passage Kant defines a schema by saying that “[the] representation of a universal procedure of imagination in providing an image for a concept, I entitle the schema of this concept” (A140/B179-180). 100 Ch. 2 Kant on Formal-Logical and Mathematical Cognition the act whereby we construct the concept, and abstract from the many determinations (for instance, the magnitudes of the sides and of the angles), which are quite indifferent, as not altering the concept ‘triangle’” (A714/B742). This contrast between the act and the figure is precisely the contrast between a “universal procedure”, i.e., a schema, and an “image for a concept”.40 To sum up the discussion so far, we have seen Kant arguing that the semantic or representational content of the concepts of geometry is given by the rules of geometrical construction. And for Kant such rules are naturally those found in Euclid’s Elements. This is the first stage in Kant’s account of the semantics of geometry. He completes this explanation by the further contention that (geometrical) con-

40 At this point a critic of Kant, registering the difference between image and schema, might question the relevance of the “figure” to geometrical proof. For example, Philip Kitcher (1975, p. 124), having observed Kant’s view that in geometrical proof “we can draw conclusions using only those features of the image on which the rule has pronounced”, writes that this seems to make the figure (the exhibition of a geometrical concept in intuition) quite unnecessary: “[f]or if all that we are allowed to do is to draw out features of triangles prescribed by the schema of the concept ‘triangle,’ then we can do this by conceptual analysis alone. [...] By resisting generalization over accidental features of the drawn figure, we seem to restrict our ability to generalize to properties which the schema demands be exhibited in all triangles. So we need only look to the schema and not to the constructed object” (ibid.) What Kitcher overlooks is the fact that, for Kant, the only way to “look to the schema” is precisely to draw a figure and perform a series of inferential steps which are “guided” by intuition. In a similar vein, Bolzano (1810, Appendix, §7) and Couturat (1904, pp. 348-352) voice the criticism that Kantian schemata, insofar as they are general rules of construction, are not distinguishable from concepts. But this criticism ignores Kant’s way of drawing the distinction between concepts and intuitions. It ignores, that is to say, his “fundamental observation” that the traditional model of concepts is insensitive to the finer propositional structures which are found in geometrical judgments and which Kant intends to capture by his notion of construction. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 101 struction qua species of representation is dependent on or grounded in the intuitive representation of space.41

2.3.2 Geometry and Space

Let us give the collective name “geometrical space” to the objects of geometrical representation. In the Doctrine of Method Kant argues that this geometrical space is “given to us” in, or represented through, construction. In the Aesthetic- and Analytic-sections the contention is found that this manner of representation and its object is grounded in space as it pertains to sensibility, or space as it is given to us prior to any determination or conceptualisation such as occurs in geometry or outer experience. This “preconceptual framework”, as Allison (1983, p. 94) calls it, is the single, infinite, subjectively given space, which is described in the Aesthetic under the heading of “Metaphysical Exposition of the Concept of Space” (A23-25/B37- 40). The objects of geometry can only be represented in intuition, by “combining” or “synthesizing” the manifold of space according to one or another rule of construction.42

41 Geometrical construction presupposes not only the intuitive representation of space but also that of time; cf. below. 42 At B137-138 Kant explains this as follows: “[T]he mere form of outer sensible intuition, space, is not yet by itself cognition; it supplies only the manifold of apriori intuition for a possible cognition. To cognize anything in space (for instance, a line) I must draw it, and thus synthetically bring into being a determinate combination of the given manifold, so that the unity of this act is at the same time the unity of consciousness (as in the concept of line); and it is through this unity of consciousness that an object (a determinate space) is first cognized.” In this passage I have substituted “cognition” and “cognize” for Kemp Smith’s “knowledge” and “know” as translations of Kant’s “Erkenntnis” and “erkennen”. That geometrical representations presuppose the synthesis of the manifold of outer intuition is made clear at B160n, where Kant writes that the representation of space as object (“as we are required to do in geometry”) involves the combination of the form of intuition in an intuitive representation, or formal intuition. For the details of the 102 Ch. 2 Kant on Formal-Logical and Mathematical Cognition

Given this dependence, it follows that geometrical representations derive their content from this original, intuitive representation of space. Or, as Kant himself puts it in his notes on Kästner, the geometer grounds the possibility of carrying out his task (geometrical construction) on space as it is originally represented.43 For instance, the geometer assumes – and this assumption is codified in the construction postulates – that straight lines can be bisected and extended indefinitely. That these constructions can be carried out presupposes space as a single, infinite (i.e., unbounded) whole, one that is divisible and extendible ad indefinitum.44 In general, then, the role of the “metaphysical space” is that of underwriting the constructive procedures of Euclid’s geometry. The attributes which Kant attaches to the “metaphysical space” are not self-explanatory. Consequently, Kant’s conception of how geometrical and metaphysical space are related to each other is not self-explanatory, either. In a recent paper on Kant’s theory of geometry, Michael Friedman has argued convincingly that the attributes are best described in kinematic terms, by referring to the idea of a subject’s imaginary motion in perceptual space (Friedman 2000, pp. 190-193). In this way we arrive at the following three explications. distinction between “form of intuition” and “formal intuition”, see Allison (1983, pp. 94-98), Buchdahl (1969, pp. 580-1). 43 Kant explains that metaphysics considers space as it is given before all determinations, whereas in geometry it is considered as it is generated; for the details of Kant’s explanation, see Allison (1973, Appendix B, pp. 175- 176). 44 This dependence-thesis is succinctly formulated by Kant’s friend and philosophical ally J. G. Schulze in his Prüfung der Kantischen Critik der reinen Vernunft: “If I should draw a line from one point to another, I must already have a space in which I can draw it. And if I am to be able to continue drawing it as long as I wish, without end, then this space must already be given to me as an unlimited one, that is, as an infinite one. Correlatively, I cannot successively generate any cylinder or body except in space, that is to say, I can only do so because this space is already given, together with its quality which allows me to suppose that points are everywhere, and which enables me to generate, without end, the three dimensions of extension” (as quoted in Allison 1983, p. 95). Ch. 2 Kant on Formal-Logical and Mathematical Cognition 103

(1) That space is single (i.e., that it is one) means that every region of space is given as a part of surrounding space; in kinematic terms, this amounts to saying that every spatial region is reach- able by a single subject’s imaginary motion in space.

(2) That space is infinite means that it is unbounded, that no region of space is given with an absolute limit;45 in kinematic terms, this amounts to saying that a spatial region is always given with the possibility of moving beyond.46

(3) That space is subjectively given means, precisely, that the space of experience is fundamentally a kinematic notion that it is given in terms of motion, “as an act of the subject” (as Kant puts it at B155).

It is clear, furthermore, that when Kant speaks of the synthesis of the manifold in space, he has this kinematic notion in mind; and also that this notion, which yields a description of space, belongs to geometry.47 In other words, he holds the view that the formal structure of

45 For Kant’s explanation, see A514-515/B542-543. The term “absolute limit” is used by Schulze in his response to Eberhard: “to say that space is infinite means that it is nowhere terminated, that no absolute limit is possible beyond which there would be no more space” (Allison 1973, p. 173). 46 Russell, referring to Kant’s view that “space is represented as an infinite given magnitude”, writes: “[t]his is the view of a person living in a flat country like that of Königsberg; I do not see how an inhabitant of an Alpine valley could adopt it. It is difficult so see how anything infinite can be ‘given.’” (1945, p. 688). After this attempt at geographic explanation Russell continues: “I should have thought it obvious that the part of space that is given is that which is peopled by objects of perception, and that for other parts we have only a feeling of possibility of motion” (ibid.). On the kinematic interpretation, this “possibility of motion” is precisely what Kant has in mind when he speaks of the infinity of space. 47 “Motion of an object in space does not belong to pure science, and consequently not to geometry. For the fact that something is movable cannot be known apriori, but only through experience. Motion, however, considered as the describing of a space, is a pure act of the successive synthesis 104 Ch. 2 Kant on Formal-Logical and Mathematical Cognition the space of our experience, which is delineated kinematically, receives its systematic expression in (Euclid’s) geometry. As Friedman (2000, p. 193) puts it, “[g]eometrical space is [...] iteratively or con- structively generated within the formal structure of perceptual space by successively applying the fundamental operations of drawing a straight line and describing a circle, and this is the precise sense [...] in which the possibility of mathematical geometry is grounded in or explained by the formal structure of perceptual space”.48, 49 of the manifold in outer intuition in general by means of the productive imagination, and belongs not only to geometry, but even to transcendental philosophy” (B155n). 48 As Friedman points out, Kant’s notion of pure intuition, when it is construed kinematically, “contains the seeds of its own destruction” (2000, p. 202). For Kant holds, in effect, that the formal structure of spatial intuition (possible spatial motions) is expressed by the conditions of free mobility and that these conditions are uniquely captured by Euclidean geometry, which is why it is apriori. After the discovery of non-Euclidean geometries, however, it became clear that these conditions do not yield the specifically Euclidean space but the three classical cases of space of constant curvature. It follows that, insofar as “spatial intuition” or “the apriori element in geometry” is tied up with free mobility, Euclidean geometry can no longer be regarded as apriori. 49 One of the distinctive virtues of the kinematic reading is that it avoids interpreting Kant’s “pure intuition” as a perception-like source of mathematical knowledge. On occasion Kant does express himself in a way which suggests that mathematical knowledge requires a special mode of cognition, apriori or pure intuition, thereby encouraging an association of his conception of that knowledge (though not its subject-matter) with what is known as mathematical platonism. For instance, at A713/B741 he writes: “[t]o construct a concept means to exhibit apriori the intuition which corresponds to the concept. For the construction of a concept we therefore need a non-empirical intuition” (italics in the original). Given that sense-perception is, if not the sole instance, then at least the paradigmatic example of empirical intuition, it begins to seem as if pure, i.e., non-empirical intuition is somehow akin to ”non- sensuous seeing”. This view is expressed in the following passage by Philip Kitcher: “pure intuition is a process in which we construct pictures in our mind’s eye, inspect them, and so arrive at knowledge of fundamental mathematical truths” (1979, p. 251). And this in turn invites an immediate comparison with a view like that Gödel expressed by his famous statement Ch. 2 Kant on Formal-Logical and Mathematical Cognition 105

2.4 Constructions in Arithmetic

Turning now from geometry to arithmetic, it must be noted, first of all, that Kant’s conception of the latter is markedly less worked out than his conception of the former. At the risk of oversimplifying a complex interpretative issue, I shall present a relatively straightforward interpretation of Kant’s theory of arithmetic. Kant’s conception of arithmetic is a representative of the so-called ratio theory of numbers. On this view, natural numbers and rational fractions are explained by referring to counting and measurement, respectively, and arithmetic emerges as a method of calculating magnitudes.50 In this connection it suffices to consider “counting numbers” or positive integers; these apply to discrete objects, and they are used to determine the magnitudes of “aggregates” or “multiplicities” of such objects. What such determination presupposes is a pre- selected unit of counting, and the determination itself consists in as- certaining how many such units the multiplicity is composed of. Thus, positive integers do not apply to discrete objects or their aggre- about our possessing “something like perception of the objects of set theory (1964, p. 483). What is missing from Gödel is of course the “constructivist” element that is present in Kant. The relevant analogy, however, does not pertain to the ontology of mathematics, but its epistemology. And here, it could be argued, Gödel’s claim that the perceptual character of (part of) set- theoretical knowledge is shown by the fact that “the axioms force themselves upon us as being true” has a predecessor in the putatively Kantian view that (pure) intuition confers immediate evidence on the propositions of mathematics (as is suggested in Parsons (1979-80)). The picture that emerges from the kinematic interpretation is clearly very different. What makes geometrical constructions intuitive is the fact that the rules underwriting them are rules for constructing individuals – objects falling under the constructed concepts. And the apriori character of mathematical knowledge is a consequence not of our possessing a special evidential faculty, but of our possessing special rules, rules which are constitutive of geometrical thinking. This is explained in more detail below, see section 2.7. 50 In a letter to Schulze, November 25, 1788, Kant explains that the object of arithmetic is quantity as such, i.e., the concept of a thing in general by means of quantitative determination (Kant’s Briefwechsel, Band I, p. 555.10-14). 106 Ch. 2 Kant on Formal-Logical and Mathematical Cognition gates simpliciter, but to what was referred to as “discrete quanta”. Kant, for his part, explains this as follows (see B??). Quantum is an aggregate or multiplicity in which the parts are homogeneous. What makes an aggregate homogeneous – allows it to be represented as one – is synthesis, whereby a manifold given in intuition is subsumed under a common concept. This is what makes quanta capable of being measured. “Number” in the sense of positive integer is thus explained as that which reflects such determination or “arises” in it. Consider in light of this brief characterisation the arithmetical example which Kant discusses at B15-16, namely the numerical formula 7 + 5 = 12. He begins by considering the suggestion – found in Leibniz – that “[w]e might [...] at first suppose that the proposition 7 + 5 = 12 is a merely analytic proposition, and follows by the principle of contradiction from the concept of the sum of 7 and 5.” Against this he argues that arithmetical addition is not an operation on concepts: “[h]owever we might turn and twist our concepts, we could never, by the mere analysis of them [...] discover what [the number is that] is the sum” (B16),51 but rests in effect on construction carried out in intuition. Arithmetical addition, that is, is an operation carried out on quanta, or multiplicities to which such numbers are assigned. Thus, Kant explains, the addition of 5 to 7 consists in the decomposition of a multiplicity of 5 objects and the addition of its elements, unit by unit, to the aggregate of 7 objects.52

51 At B205 Kant argues that the representation of the number 12 is not contained in the representation of the number 7 or that of 5 nor in the representation of their combination (Zusammensetzung). By “combination” he evidently has in mind what we should expect by now, to wit, the composition of complex concepts out of their marks; see Longuenesse (1998, pp. 277-278). 52 Recall that, according to Leibniz, the proof of the proposition 2 + 2 = 4 is the sequence 4 = 4; 4 = 3 + 1; 4 = 2 + 1 + 1; 4 = 2 + 2, grounded in the substitution of definitionally equivalent terms. As Longuenesse (1998, pp. 282) points out, Kant’s argument against this construal of the proof- ground rests on his view that the definitions (4 =df 3 + 1, 3 =df 2 + 1, etc.) as well as the substitution are themselves based on certain synthetic operations. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 107

The distinction between images and schemata that we envisaged in connection with Kant’s theory of geometry applies to arithmetic as well. Any multiplicity to which a number applies (any quantum) serves as an image of that number (cf. Kant’s explanation at A140- 141/B179-180). The schema, by contrast, is to be found in the general procedure underlying any such calculation: “the pure schema of magnitude as a concept of understanding is number, which is a representation that unites the successive addition of unit to (homogeneous) unit” (A142/B182). And it is the necessary temporal character of this schema that makes the concept of number a temporal one: “[t]he concept of magnitude in general can never be explained except by saying that it is that determination of a thing whereby we are enabled to think how many times a unit is posited in it. But this how-many- times is based on successive repetition, and therefore on time and the synthesis of the homogeneous in time.” (A241/B300). This explanation, it is to be noted, makes no mention of the specific rules which govern arithmetical operations. In the case of geometry, it follows from the necessarily temporal character of the concept of magnitude that this science, too, presupposes the intuitive representation of time; as we saw above, the subject-matter of geometry is generated by the iterated application of the construction postulates to the intuitive manifold of space. However, it was also argued that, for Kant, the rules of geometrical construction derive their specifically geometrical content from the original, intuitive representation of space. In the case of arithmetic, by contrast, there is clearly no such dependence of the rules of arithmetic on time; for instance, the associativity of addition cannot be grounded in the properties of time in the way that the infinite extendability of line segments was derived from the unboundedness of perceptual space. It would seem therefore that the relation that arithmetic bears upon time is due to the necessarily temporal nature of the concept of magnitude, i.e., their being constructible by an iterative, step-by-step, procedure.53

53 It follows from this relation of arithmetic and time that the science itself should be assigned to the understanding rather than sensibility. That this 108 Ch. 2 Kant on Formal-Logical and Mathematical Cognition

2.5 The Applicability of Mathematics

So far I have given on overview of Kant’s representation-theoretic views as they appear in his theory of geometry and arithmetic.54 They form the gist of his conception of mathematics as synthetic. They also enable him to answer what was for him probably the philosophically most urgent question concerning mathematics, to wit, its applicability. Here, again, the relevant contrast is between Kant, on one hand, and Leibniz and his followers, on the other. In Kant’s view, the rationalist conception of mathematics was particularly prone to generate sceptical doubt. The reasons for this are described succinctly by J. Michael Young: “[b]ecause mathematics is a body of apriori knowledge, and because its concepts are pure, Kant’s rationalist predeces- is Kant’s view is made clear in his letter to Schultz. There he says, admitting a point made by his correspondent, that time has no influence upon the properties of numbers. For this reason “the science of number” is grounded in “a purely intellectual synthesis, which we represent to ourselves in thought” (Kant’s Briefwechsel, Band I, p. 557.3-6). 54 Kant’s theory of mathematics covers not only geometry and arithmetic, but extends to analysis and algebra as well. His conception of the generation of magnitudes is not dissimilar to Newton views (and is arguably influenced by them); what Kant says about the calculus finds a natural ally in Newton’s calculus of fluxions, which makes essential use of temporal and kinematic notions (for some of the details, see Friedman 1992a, pp. 71-80). For Kant’s views on algebra, see Shabel (2003, sec. 3.3). She shows that Kant’s brief references to algebra and “symbolic construction” are best understood by connecting them to the 18th century tradition which regarded algebra as a method for solving arithmetic and geometric problems. In the wake of Descartes’ and Fermat’s work on analytic geometry it became possible to replace compass and straightedge constructions with equations for these curves. In the course of the 18th century, notable mathematicians like Euler and Lagrange began to pursue the “analytic method” on its own, recognising its superiority to the “synthetic method”. Kant, whose conception of geometry is firmly rooted in the traditional, synthetic method of straightedge and compass construction, naturally regards algebra (“symbolic construction”) in the less ambitious manner, as a shorthand for manipulating geometrically constructible magnitudes, which can be exhibited in intuition (“ostensive construction”). Ch. 2 Kant on Formal-Logical and Mathematical Cognition 109 sors were lulled into believing that mathematical concepts are not tied in any essential way to sensible intuition. Ironically, this belief subse- quently bred skeptical doubt as to whether mathematical concepts are applicable to the objects we perceive, and whether mathematics is therefore true of the empirical world” (1994, p. 353). Kant’s response to this problem can be described as follows (here I will use the case of geometry as an illustration, as does Kant himself at B205, where he explains the essentials of his response). In his view, space has two roles to play. Firstly, “[e]mpirical intuition is possible only by means of the pure intuition of space” (ibid.) Secondly, the pure intuition of space is presupposed in geometry, since it is what first gives geometrical concepts their content (this is Kant’s representation-theoretic thesis). Thus, on one hand, space is constitutive of our experience; on the other hand, this form, which enables us to cognize objects as possessing spatial magnitudes, is also presupposed in geometry. It follows that even though geometry, considered in itself, yields only what Kant calls the form of appearances or empirical objects (“through the determination of pure intuition we can acquire apriori knowledge of objects, as in mathematics, but only in regard of their form” (B147)), this form is nevertheless necessarily applicable to empirical objects.55

2.6 Pure and Applied Mathematics

For our purposes the importance of Kant’s anti-sceptical argument lies in the fact that it puts us in a position to consider an issue which has played a major role in later discussions of his theory of mathematics, Russell’s being a prominent early example. This issue is the distinction between pure and applied mathematics. One of the standard criticisms which 20th century philosophers – logical empiricists in particular – levelled against that theory is that it ignores the crucial differences between these two ways of looking at a mathematical theory.

55 Cf. here A223-224/B271 and Prolegomena, Note I to the First Part of the Main Transcendental Question. 110 Ch. 2 Kant on Formal-Logical and Mathematical Cognition

Even though philosophers and logicians have suggested more than one explanation of what “pure mathematics” is, the distinction pure vs. applied itself was often taken to undermine Kant’s view that mathematical truths are both synthetic and apriori. For instance, geometry as a branch of pure mathematics is indeed apriori, but this is taken to follow precisely from its purity, i.e., from the fact that it is independent of all questions concerning empirical applicability. By contrast, statements of applied geometry are said to be synthetic in the sense that they describe the geometrical structure of physical space; what this structure is like, however, is at least partly a matter of empirical investigation and therefore cannot be determined apriori.56 Kant, however, does draw a distinction between “pure” and “applied” mathematics. Or, to put it more cautiously, there are elements in his theory of mathematics which, when combined with the distinctively Kantian conception of human cognition, suggest the possibility of envisaging mathematics independently of the problems which concerning its empirical applicability. The suggestion, to put it very briefly, is this. The sphere of pure mathematics in the Kantian sense is the sphere of pure intuition regardless of the problem – raised and answered in transcendental philosophy – concerning the so-called real possibility of objects conforming to the dictates of pure intuition (concerning the objective validity of these dictates); even though pure intuition delivers only the form of appearances, this form can itself be

56 The later Russell puts the point as follows: “‘Geometry’, as we now know, is a name covering two different studies. On the one hand, there is pure geometry, which deduces consequences from axioms, without inquiring whether the axioms are ‘true’; this contains nothing that does not follow from logic, and is not ‘synthetic’, and has no need of figures such as are used in geometrical text-books. On the other hand, there is geometry as a branch of physics, as it appears, for example, in the general theory of relativity; this is an empirical science, in which the axioms are inferred from measurements, and are found to differ from Euclid’s. Thus of the two kinds of geometry one is apriori but not synthetic, while the other is synthetic but not apriori. This disposes of the transcendental argument” (1945, p. 743). Ch. 2 Kant on Formal-Logical and Mathematical Cognition 111 regarded as an object of study, and is so regarded by the mathematician.57 Even in the Kantian context, then, there is room for distinguishing mathematical thought as such (pure mathematics) and the question of its empirical applicability. To be sure, the Kantian way of differentiating between the two is altogether different from its more modern articulations (whatever form, exactly, these may take). Kant’s representation-theoretic thesis effectively excludes the point of view of “abstract” mathematics; even at the level of pure mathematics, mathematical concepts cannot be divested from the constructibility and its presuppositions, since this is the only way to represent these

57 This description needs a qualification. It applies best to geometry, which has a domain of objects as its subject-matter, to wit, the geometrical construction which collectively constitute “geometrical space”. Since “metaphysical space” is a form of empirical objects, these constructions, being grounded in the metaphysical space, are describable as “forms of objects” from the standpoint of transcendental philosophy; nevertheless, there is a perfectly respectable sense in which they are objects, since they are so regarded by the geometer. Geometry can thus be said to be a science whose objects are the constructed spatial magnitudes. “Pure” arithmetic and algebra, on the other hand, are more abstract than geometry. They are not about any objects î not even about such quasi-objects as geometry î but deal with the mere concept of magnitude, or more precisely, with the procedure of constructing or calculating any magnitude: “But mathematics does not only construct magnitudes (quanta) as in geometry; it also constructs magnitude as such (quantitas), as in algebra. In this it abstracts completely from the properties of the object that is to be thought in terms of such a concept of magnitude. It then chooses a certain notation for all constructions of magnitude as such (numbers), that is for addition, subtraction, extraction of roots, etc. Once it has adopted a notation for the general concept of magnitudes so far as their different relations are concerned, it exhibits in intuition, in accordance with certain universal rules, all the various operations through which the magnitudes are produced and modified” (A717/B745). This difference between geometry, on one hand, and arithmetic and algebra, on the other, does not undermine the Kantian distinction between pure and applied mathematics; quite clearly, the general rules for constructing “magnitude as such” can be studies independently of their application to some empirical realm. 112 Ch. 2 Kant on Formal-Logical and Mathematical Cognition concepts in a manner which is a necessary prerequisite for their having the sort of content and function that they do have in mathematical theories.

2.7. The Apriority of Mathematics, According to Kant

Let us now turn to consider the epistemic component of Kant’ apriorist program, that is, his account of why mathematical truths are apriori knowable. Given the distinction between pure and applied mathematics, these explanations will differ, depending on whether we consider mathematical judgments on their own right or envisage them as applied to empirical objects. Note, first, the following. On Kant’s view, mathematical thought establishes judgments like

(#) Necessarily, every triangle has the angle-sum property.

That the judgments delivered by pure mathematics or mathematical thought have this modal status follows straightaway from Kant’s explanation of their content. This connection has been convincingly explained by Michael Friedman (1992a, pp. 127-8). For example, the only way we can represent the angle-sum property is by carrying out a required construction (in this case Euclid’s proposition I.32). It automatically follows that any object falling under the concept triangle has the angle-sum property; a triangle not possessing it is therefore not really a genuine possibility at all. It is worthwhile to spell out in some detail the senses of possibility and impossibility that are at play here. It follows from Kant’s account of mathematical necessity that the impossibility of a non- constructible mathematical concept is a matter of mathematical thought as such (pure mathematics) and not of its application to appearances. Mathematical concepts must be constructible, not because constructibility is needed to forge a link between these concepts and appearances, but because construction is necessary for the very possibility of mathematical thought itself. Hence, the sense that attaches Ch. 2 Kant on Formal-Logical and Mathematical Cognition 113 to the necessity-operator in the judgment (#) is the strongest possible sense of necessity: there is no sense in which the negation of what lies in the scope of the modal operator is possible.58 This point can also be formulated as follows. Kant’s picture of mathematical necessity implies that mathematical existence is coextensive with constructibility: the subject-matter of mathematics consists of certain “mental operations”, namely constructions, and mathematical necessity is a matter of what necessarily follows from such constructions. Without this assumption, there clearly is no valid inference from “P is a property which necessarily belongs to every constructible triangle” to “every triangle is necessarily P”. Consider, next, the question of apriority. Again, as Michael Friedman (1992a, p. 127) explains, the bulk of explanation is carried by Kant’s conception of the content of mathematical judgment. He formulates the point as follows: “The truth of the true propositions of mathematics follows from the mere possibility of thinking or rep-

58 The alternative interpretation has been well expressed by Gottfried Martin (1955); (1) mathematics is intuitive, according to Kant, because it “is limited to objects which can be constructed” (p. 23); therefore (2) intuition “limits” the broader region of logical existence [...] to the narrower region of mathematical existence” (p. 25); thus, (3) mathematical theories like non- Euclidean geometries “are logically possible but they cannot be constructed; hence they have no mathematical existence for Kant and are mere figments of thought” (p. 24; the allusion is to A220-1/B267-8). This view can also be expressed in terms of possible worlds (cf. here Brittan (1978, pp. 13-28 and Kitcher (1975, 110-3)): Martin’s “logical existence” refers to “logically possible worlds”, from which “real” or “mathematical existence” singles out a sub-class, roughly, worlds which are not only logically but also experientally- cum-mathematically possible. On the alternative view formulated by Fried- man, the relation between these two notions of possibility is subtler (as Friedman (1992a, p. 128) himself observes). In particular, Kant’s notion of “real possibility” is exactly on a par, as regards its modal strength, to our notion of logical possibility: “[a]lthough concepts that are mathematically impossible can indeed be logically consistent [according to Kant], it does not follow that there are logically possible states of affairs corresponding to such concepts” (ibid.) 114 Ch. 2 Kant on Formal-Logical and Mathematical Cognition resenting them, and this is the precise sense in which we know them apriori” (1992, p. 127). This particular formulation is not entirely satisfactory, however. As it stands, it characterises the necessity of mathematics rather than its apriority. To elaborate on this, I shall consider some of the points that Philip Kitcher (1986, sec. 4) makes about the notion of apriority. Kitcher argues that the traditional notion of apriority covers more than one notion. Firstly, there is the familiar idea of apriority as being concerned with epistemic warrant; very roughly, a proposition is apriori in this sense (“aprioriW”), that is to say, is apriori knowable or has an apriori warrant, only if it can be known independently of experience. Secondly, there is what we could call the presuppositional sense of apriori (“aprioriP”); a proposition or a concept is aprioriP only if it is a prerequisite for thought or experience (of some specific kind or generally). Now, Kitcher is principally concerned with showing that these two conceptions, aprioriw and aprioriP, are distinct. Consider, for instance, the traditional view that the law of (non-)contradiction is a “law of thought”. This claim was often defended by arguing that if this law is given up, then rational thought and discourse are no longer possible (id., p. 314). This defence, Kitcher points out, has no bearing on whether the law is aprioriW. Even though Kitcher’s specific claims about their interconnections may not stand critical scrutiny,59 it seems correct to say that the two notions are, indeed, distinct. In particular, identifying a proposition as aprioriP (or as a necessary consequence of a proposition which is aprioriP) does not, in itself, say anything about the question of epistemic warrant. Returning now to Friedman’s characterisation of the Kantian apriority, we can see that the notion he has in mind is essentially Kitcher’s aprioriP. That this is part of what Kant meant by apriority is clear enough.60 Equally clearly, he often considers apriority as a characteristic of items of knowledge and what he then has in mind is

59 Cf. here Wagner (1992, sec. V). 60 To take a random example, Kant argues in the Aesthetic that space is an apriori representation which underlies all outer intuitions as their condition of possibility (A24/B38). Ch. 2 Kant on Formal-Logical and Mathematical Cognition 115

Kitcher’s aprioriw.61 Consequently, Friedman’s explanation does not, as such, cover the whole field of the Kantian apriori. The relevant extension, however, is readily available once we pay attention to one further feature of Kant’s aprioriP which Friedman, as far as I can see, fails to make explicit in his explanation. In order to see how the gap between Kant’s aprioriP and aprioriW can be closed, we should take notice of the fact that what according to Kant is a method of mathematical representation, namely construction, is at the same time the method of obtaining mathematical knowledge. For example, the rigorous representation of a geometrical figure is necessarily tied up with its construction; but it is precisely by means of this method that we attain substantial knowledge of geometrical objects; we could say that, for Kant, there is in mathematics no difference between thinking and knowing.62 Kant himself says this much at A88/B121: “[t]he objects, so far as their form is concerned, are given, through the very knowledge [Erkenntnis] of them, a priori.”63 That is to say, the geometrical-cum-spatial forms of empirical objects can only be represented through geometrical construction, which is therefore the prerequisite for geometrical thought. At the same time, this very same procedure confers an apriori warrant on the geometrical propositions which result from such construction, since this procedure is a necessary ingredient in the Euclid’s proof of that proposition.64

61 To take another random example, Kant argues in Prolegomena, §2b, that the analytic judgment “gold is a yellow metal” is apriori, for to know this “I need no further experience outside my concept of gold, which contained that this body is yellow and metal”. 62 I borrow this formulation from Ferrarin (1995, p. 136): “Since the mathematician constructs or exhibits his object in an apriori intuition, thinking and knowing are not separate in mathematics”. Ferrarin does not, however, give a detailed explanation of how this connection arises in the Kantian context. 63 Here it is appropriate to translate Erkenntnis as “knowledge” since Kant’s point in the passage is that geometrical Erkenntnis possesses “immediate evidence”, thanks to its grounding in the pure intuition of space. 64 Again, this “cannot be represented” must be read in an ontologically pregnant way: triangles exist only as potential objects of mathematical operations (“they are given through the very knowledge of them”). For otherwise 116 Ch. 2 Kant on Formal-Logical and Mathematical Cognition

Not only pure but also applied mathematics is apriori knowable, according to Kant. His argument for this conclusion can be formulated as follows:

(1) we know apriori that there cannot be constructible triangles which do not have the angle-sum property; (2) the act of construction which confers an apriori warrant on (1) is the same act which is involved in the apprehension of empirical objects; (3) therefore, we know apriori that there cannot be empirical triangles which do not have the angle-sum property.

Kant’s argument for the apriori knowability of mathematics is thus based on the assumption that construction is apriorip not only with respect to pure mathematics but also with respect to applied mathematics. That is, he holds that mathematics is a necessary ingredient in the apprehension of empirical objects. The results established by mathematical thought are thus (necessarily) transferable to appearances. As Kant himself emphasised, this argument has far-reaching consequences.65 These are, in fact, precisely analogous to those that we encountered in connection with pure mathematics. In general, mathematical judgments can be regarded as necessary and apriori knowable only on the assumption that their subject-matter depends for its existence on mathematical thought. Since mathematical one might argue that, besides those triangles that can be represented, there are others which cannot, and these may or may not have the property. To be sure, these non-constructible triangles would not fall under the concept triangle, but this would be of no consequence, since there is no general dependence between how things are and the concepts that we make to ourselves of these things. We would not, then, know apriori that all triangles have the angle-sum property. That Kant was moved by this kind of “realist” argument is shown by his well-known inference of transcendental idealism from the apriority of applied mathematical knowledge, but similar considerations apply to pure mathematics as well. 65 See, e.g., Prolegomena, §§ 8-11, Transcendental Aesthetic, A47-9/B64-6. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 117 thought is grounded in the spatio-temporal activity of construction, and this activity is presupposed in the apprehension of empirical objects, space and time themselves can be attributed to these objects only insofar as these are regarded as appearances and not as things in themselves. This, in brief, is Kant’s argument from the nature of mathematical knowledge to transcendental idealism.

2.8 Conclusions

In this chapter I have given an outline of a contextualist interpretation of Kant’s theory of mathematics. That is to say, the starting- point of what I have called the “semantic interpretation” is the view that Kant’s conception of mathematical method was tied up with certain eighteenth century mathematical practices. This knowledge, moreover, was derived not so much from an actual confrontation with those practices themselves as from standard German textbooks on mathematics.66 More precisely, the Kantian version of the apriorist program was built, or arose from a reflection, on the geometrical approach to mathematics which had dominated mathematical thought from its Greek beginning to the early seventeenth century, or thereabouts, and which, apparently, was still dominating more popular expositions of mathematical ideas.67 It is this approach the essentials of which Kant tried to capture with the help of the notion of construction in pure intuition; this notion, he argued, is crucial for a proper understanding of the content of mathematical judgments, the character of mathematical reasoning and, in general, the method underwriting mathematical knowledge. What Kant was not in a position to see was that, already by the end of the eighteenth century, the geometrical approach had lost all its credentials among mathematicians. Analysis had become the most active branch of mathematics in the eighteenth century, and the ex-

66 Cf. here Shabel (2003). 67 For a brief characterisation of the geometrical approach, see, e.g., Kline (1990, Ch. 18). 118 Ch. 2 Kant on Formal-Logical and Mathematical Cognition tensions introduced therein, which eventually gave rise to whole new branches of mathematics, were effected by “purely analytical means”, to wit, by replacing the earlier geometric-cum-intuitive semantics by analytical expressions or functional equations and their manipulation in accordance with the rules of analysis.68 To be sure, the meaning which eighteenth century mathematicians associated with this manipulation of symbols was very far from what such a phrase came to mean in the hands of those mathematicians and logicians who eventually logicized mathematical reasoning. There was no “logic” underlying this manipulation of symbols or derivation of equations, and the rules themselves were trusted primarily because the calculus proved highly successful in solving problems in physics. Thus, mathematical reasoning did retain its intuitive character, and potential complaints about missing rigour were put aside as unnecessary worries which merely stood in the way of mathematical progress (cf. Kline 1990, pp. 614-6). This change in general perspective, i.e., the substitution of algebraic analysis for the geometrical approach, generated a whole host of foundational problems concerning the semantics and methodology of mathematics, questions which the pragmatically minded mathematicians did not immediately appreciate. Although the series of nineteenth century developments which came to be known as the pursuit of rigour was arguably dictated by the needs of mathematical research (Kitcher 1983, p. 268) and not by any deeply felt sense of foundational crisis, it nevertheless resulted in a new conception of foundations which was totally different from, and stood in a sharp contrast with, that underlying the geometrical approach. As regards philosophy of mathematics, these nineteenth century developments were anything but congenial to a Kantian construal of mathematics. In particular, they undermined very effectively the se-

68 Similar development took place in geometry, where the synthetic method, operating on drawn figures, gave way to an analytic or algebraic approach. Again, the primary reason operative in this change was the observation that the new language promised extensions in mathematical knowledge which were beyond the purview of the traditional, synthetic method. Ch. 2 Kant on Formal-Logical and Mathematical Cognition 119 mantic connection between mathematical sciences and the apriori intuitions of space and time which underlie Kant’s theory of mathematics. Similarly, the new picture of mathematics, when formulated explicitly towards the end of the century, provided entirely new standards for evaluating the rigour or otherwise of mathematical theoris- ing. When they measured Kant’s theory by the new standards, many – though by no means all – mathematicians and philosophers concluded that the transformation of logic and mathematics had deprived Kant’s doctrines of all plausibility. This is the starting-point of Russell’s criticisms of Kant.

Chapter 3 Russell on Kant

3.0 Introduction

In chapter 1 I gave a preliminary description of the motives behind Russell’s early logicism. I argued that Russell’s reasons for insisting on rigour were not epistemological but, broadly speaking, semantic. Logi- cism, that is, was not motivated by sceptical worries, or a search for a secure and unshakeable foundation for mathematical knowledge. Rather, it was the position that Russell eventually arrived at in an attempt to gain, in Coffa’s words, “a clear account of the basic notions of a discipline”. In Russell’s case, the relevant disciplines are the different branches of pure mathematics, and to gain a clear account of their basic notions is to explain, in a manner that accords with “modern mathematics”, what is really involved in the concepts, propositions and reasonings of pure mathematics. Construed in this way, the questions with which the logicist Russell is concerned are the very same representation-theoretic or semantic questions that Kant had dealt with in the semantic component of his apriorist programme. In chapter 2 we saw Kant arguing that an explanation of the mathematical apriori must be rendered consistent with the true mathematical method, a requirement which existing models of apriority failed to meet. To fill this gap, he introduced the notion of pure intuition, or construction of concepts in spatial and/or temporal intuition. This is the cornerstone of Kantian proto-semantics for mathematics. For it is by means of construction that the mathematician gets beyond what is contained in a concept, when the latter is thought of as nothing more than a combination of characteristics. Thus, mathematical concepts acquire their specifically mathematical content only through their connection with intuition. For much the same reason, construction or intuitive representation is also a necessary ingredient in mathematical reasoning. The logicist Russell was sharply critical of Kantian semantics and the resulting theory of mathematics. As he saw it, this theory received 122 Ch. 3 Russell on Kant its sole justification from the peculiarities of a certain mathematical practice, namely the geometrical approach, which Kant had mistaken for the essence of the true method of mathematics. When the relevant mathematical and logical facts were set straight, the result was a picture of pure mathematics which rendered the geometrical approach obsolete. Consequently, Russell argued, Kant’s attempt to give a unified account of the mathematical method with the help of the notion of pure intuition had lost its explanatory power. The emphasis on rigour thus requires that a new answer be given to the question as to the nature of pure mathematics. The explanatory roles which Kant had assigned to pure intuition are now taken over by mathematical logic, considered somewhat in the manner of Leibniz. That is, logic is seen as constituting both an ideal language (a Leib- nizian lingua universalis or characteristica universalis) for the representation of mathematical content and a deductive system, a calculus ratiocinator, which is intended to capture the notion of correct mathematical inference. In this way, once its scope is appropriately extended, formal logic – a discipline not wholly unrelated to what Kant called pure general logic – promises precisely that kind of insight into the content of mathematical propositions and the nature of mathematical reasoning which Kant had denied to pure general logic. The promised insight is articulated by reconstructing the propositions and chains of inference connecting those propositions that together constitute a branch of pure mathematics. It is this method of logical reconstruction which Rus- sell refers to as analysis. In the case of mathematics we may distinguish two aspects in it. Firstly, there is the idea of analysis as rigorous axiomatisation, or the exhibition of the relevant subject-matter in terms of a number of primitives, i.e., axioms and undefined concepts, from which the rest of the subject-matter is derived, i.e., concepts definable by means of the primitive notions of the theory and theorems deduced purely logically from the axioms and definitions. Secondly, there is the idea of analysis as reduction. For Russell argues that once the deductive development of pure mathematics is pursued to its limit, logicism follows: “all pure mathematics deals exclusively with concepts definable in terms of a very small number of logical con- Ch. 3 Russell on Kant 123 cepts, and [...] all its propositions are deducible from a very small number of fundamental logical principles” (1903a, p. xv). The logicist reconstruction of pure mathematics forms the gist of Russell’s criticism of Kant: logicism provides the conclusive demonstration that the concepts, propositions and proofs of pure mathematics can be fully captured without assuming pure intuition. This reconstruction together with the underlying conception of logic is then put to use in a set of arguments against Kant’s transcendental idealism.1 As we saw in chapter 1, the logicist Russell does not deny the syntheticity of mathematics; in fact, he endorses this view. But the substitution of logic for pure intuition implies an entirely new account of the synthetic apriori. The implications of the view that logic is synthetic will be explored later (see chapter 5). In this chapter I will focus on the notion of apriori. Kant’s explanation of how there can be synthetic judgments that are nevertheless apriori commits him to transcendental idealism. This is the view that the properties that propositions possess in virtue of their apriority can be understood only on the assumption that their source is in the way these propositions are cognized. Russell considers this a major defect in Kant’s philosophy, arguing Kant’s theory of mathematics – his explanation of how mathematical judgments can be synthetic and apriori – is defective in that it misrepresents or compromises the senses of those attributes which it (correctly) ascribes to the propositions of mathematics: it does not allow them to be true; it does not allow them to be universal and it does not make them genuinely necessary. In other words, it misrepresents the sense in which these propositions are apriori. Russell argues that logicism does better than Kant’s theory in explaining the attribution of these characteristics to mathematics. A failure in this respect is one that he sees as constituting a powerful argument against transcendental idealism.

1 Russell’s arguments are intended to apply not only to Kant’s transcendental idealism but to “idealism” of any sort and kind. In what follows, my primary focus will be on the case of Kant, however. 124 Ch. 3 Russell on Kant

3.1 Russell on the Nature of the Mathematical Method

3.1.1 “The Most Important Year in my Intellectual Life”

Russell had been occupied with the philosophy of mathematics from the very beginning of his philosophical career. His first attempts were marred by an acceptance of certain idealist dogmas.2 Having been an idealist, he turned to a rather extreme form of realism, describable, perhaps, as a version of platonism. In this development he was aided by G. E. Moore, although most of the details had to be worked out by Russell himself.3 Moorean metaphysics and logic, however, did not produce much progress in the analysis of mathematical reasoning. It was only when Russell learned about Peano’s mathematical logic that he felt he had found an adequate method which made genuine progress possible.4 “The most important year in my intellectual life,” Russell wrote in retrospect, “was the year 1900, and the most important event in this year was my visit to the International Congress in Philosophy in Paris” (1944, p. 12).5 There was at the congress “much first-rate dis-

2 Griffin (1991) includes a detailed discussion of Russell reasons for rejecting idealism. 3 See, again, Griffin (1991) for details. Some of these are discussed below, in section 4.4, where the focus will be on the Moorean-Russellian notion of proposition. 4 Russell’s encounter with Peano was not the first time he learned about “symbolic logic”. He was familiar with the Boolean tradition, both in its original form and with the emendations introduced to it later by Peirce and Schröder. His judgment, however, was negative: “Until I got hold of Peano, it had never struck me that Symbolic Logic would be of any use for the Principles of mathematics, because I knew the Boolian stuff and found it useless” (letter to Jourdain, 15 April, 1911; quoted in Grattan-Guinness 1977, p. 133). 5 See Grattan-Guinness (1996-97) for details. Grattan-Guinness conjec- tures that the most important event in the most important year of Russell intellectual development in fact took place on 10 AM, Friday, August the 3rd, Ch. 3 Russell on Kant 125 cussion of mathematical philosophy”, he wrote to Moore immediately after his return from Paris.6 What impressed him most was Peano, whose argumentative superiority Russell attributed to his mathematical logic. Having mastered Peano’s notation, Russell found that it “afforded an instrument of logical analysis” such as he “had been looking for years” and that by studying the work of the Italian mathematicians he was acquiring a “new and powerful technique” for the work in mathematical philosophy that he wanted to do (1967, p. 144).

3.1.2 Russell and Leibniz

The idea of mathematical logic as a genuine analytical tool, or as the method of philosophical analysis, is one that Russell got from Peano, but an important ideological background was provided by Leibniz. Russell was obviously much impressed by the prospects of seeing modern mathematics with its pursuit of rigour as a partial realization of Leibniz’s ambitious programme of a scientia generalis, or a universal science, i.e., logic as the expression of the principles in accordance with which all human knowledge could be developed into a body of demonstrated truths.7 Even though he regarded Leibniz’s programme as too ambitious, or its underlying idea as misconceived, he thought it did have one correct and extremely important application, namely pure mathematics: “Leibniz’s panlogism, his belief in the possibility of deducing everything à priori from a small number of premisses, led him to conceive all truth as an ordered chain of deduction in a sense which is essentially false. In Pure Mathematics, where alone this idea is applicable, the task which he attempted has been at last accom-

as it was then that Peano’s read his paper on definitions on mathematics that made such a huge impression on Russell. 6 Russell’s letter to Moore on 16 August, 1900 (Russell 1992b, p. 202). 7 See Kauppi (1969) for further discussion. 126 Ch. 3 Russell on Kant plished” (1903b, p. 541); there, he thought, “Leibniz’s dream has become sober fact” (1901a, p. 369). Russell rejected not only panlogism; he was also sharply critical of the principles constituting Leibniz’s universal science. As one would expect, Russell argued that Leibniz’s fundamental error was his dependence on traditional subject-predicate logic, which received its expression in the “analytical theory of judgment” or “analytical theory of truth” (Russell 1903b, p. 542). This theory provided the basis for Leibniz’s notion of deductive proof and axiomatic theories, i.e., for his views on logical grammar and the presentation of mathematical truth as an “ordered chain of deduction”. Because the underlying theory was radically mistaken, Leibniz’s specific suggestions had to be rejected before any genuine progress was possible, however. In spite of these differences, Russell thought that Leibniz and modern mathematics formed a powerful alliance.8 In particular, the latter had confirmed Leibniz’s conviction that there is no such thing as a distinctively mathematical method: “[a]ll certain knowledge, Leibniz says, incorporates logical forms” (Russell 1903b, p. 552), and this insight holds good even after it is recognised that the stock of logical forms which Leibniz had acknowledged is insufficient for the task he had set to himself. Thus, Russell found himself in a position to state that “it is now known, with all the certainty of the multiplica- tion-table that Leibniz is in the right and Kant in the wrong on this point” (ibid.; emphasis in the original). Russell’s Leibnizian background consists in the acceptance of the following two views. The first may be termed a global conception of logic. According to this view, all thought and reasoning has, irrespective of its subject-matter, a common logical structure to it. Logic, in other

8 Leibniz’s influence on Russell is thus best described as ideological or programmatic in character, a point which is clearly expressed by Russell himself in a letter to Couturat, written on 9 June 1903: “What was most important in Leibniz, it seems to me, is not what he accomplished, but rather the character of the goals which he set for himself – encyclopedic, rational, deductive, systematic” (Russell 1994, p. 536). Ch. 3 Russell on Kant 127 words, is regarded as exhaustive of the principles of correct reasoning and correct representation.9 This conception goes directly against Kant, according to whom mathematical thought and reasoning are grounded, precisely, in the local or topic-sensitive rules that govern the apriori construction of the concept of quantity. The second view, Leibniz’s programme, concerns the application of the global conception of logic to mathematics. The topic-neutral or general logic is the method of mathematics, a claim that is put to work, and the correctness of which is demonstrated, by setting up an artificial language which permits the perspicuous representation of mathematical propositions and reasoning in a manner which shows what is really involved in the idea of mathematics as a genuinely demonstrative science.

3.1.3 Russell and Peano

The actual implementation of Leibniz’s programme was left to Peano and his school.10 Peano himself regarded his own work in mathemati-

9 According to Leibniz, the primary application of the principles and laws of logic is not to thinking or reasoning or representing, i.e., certain kinds of mental operations, but their objects, or truths in the natural, objective order. This objective, logical order must be kept distinct from the genealogical or historical order, which can only keep track of the way in which someone has reached some particular truth or has come to possess whatever knowledge he or she has. The genealogical order is necessarily subjective in the sense of varying from one person or community to another. By contrast, the natural order is objective, at least potentially the same for everyone; it has to do not with the actual acquisition of knowledge but with the structure of objective dependencies between truths. These are what they are independently of whether or not anyone recognizes them as such; see New Es- says, bk. IV, Ch. vii, §9; bk. iv, Ch. xvii, §3). This view or something close to it is a familiar theme in such later philosophers as Bolzano, Frege and Rus- sell, who shared a common Leibnizian “ideology”. 10 “Two hundred years ago, Leibniz foresaw the science which Peano has perfected, and endeavoured to create it. He was prevented from succeeding 128 Ch. 3 Russell on Kant cal logic as a continuation of Leibniz’s efforts,11 and he laid great emphasis on the benefits that would accrue to mathematical thinking from the translation of the different branches of mathematics into an artificial symbolic language specially designed for that purpose. It has been suggested that Peano’s reasons for promoting an ideography were predominantly pragmatic.12 Such a view, though, must be bal- by respect for the authority of Aristotle, whom he could not believe guilty of definite formal fallacies” (1901a, p. 369). In addition to Peano, Russell should have referred to Frege, who described his own concept-script in Leibnizian terms (see, e.g., Frege 1897). Russell’s first contact with Frege’s work took place when he received a copy of Begriffsschrift from his teacher James Ward (Russell 1967, p. 68). Exactly when this happened is unknown, but by the autumn of 1900 he was sufficiently familiar with Frege’s work to be able to write that Frege had “acquired almost none of the great credit he deserves” (1901b, p. 352, n1). It is commonly thought, however, that he did not begin a serious reading of Frege until the summer of 1902, when the manuscript of the Principles had already been sent to the publisher (Grattan- Guinness 1996-97, p. 125). This view seems to be confirmed by Russell’s first letter to Frege (Frege 1980, p. 130). 11 “Leibniz stated two centuries ago the project of creating a universal writing in which all composite ideas would be expressed by means of conventional signs for simple ideas, according to fixed rules” (Peano, Notations de logique mathématique (Introduction au Formulaire de Mathématiques) (1894)); quoted in Kennedy 1980, p. 46). “This dream,” Peano wrote elsewhere, “has become a reality [...] We now have the solution to the problem proposed by Leibniz. I say ‘the solution’ and not ‘a solution’, for it is unique. Mathemati- cal logic, the new science resulting from this research, has for its object the properties of the operations and relations of logic. Its object, then, is a set of truths, not conventions” (from the Introduction to the second volume of Formulaire de Mathématiques; quoted in Kennedy 1980, pp. 65-66). Peano and his disciples applied the new science of mathematical logic to different branches of mathematics. These applications were published in five subsequent volumes. The first three appeared under the title of Formulaire de Mathématiques (1895, 1897-99, 1901), the fourth was entitled Formulaire mathematique (1903), and the last, in which Peano’s Latino sine flexione was used, was entitled Formulario mathematico (1908). 12 “Peano was interested primarily in the clear presentation of mathematical results, and it was for this purpose, rather than for philosophical Ch. 3 Russell on Kant 129 anced by clear statements on Peano’s part that his notation is intended as a contribution to the clarification of the foundations of mathematics.13 Like Leibniz’s universal characteristic, Peano’s ide- analysis that he composed his Notations de logique mathématique of 1894” (Kneale & Kneale 1962, p. 473). A similar view is expressed in Frege (1897). At least some of the descriptions which Peano gives of the benefits that attach to his conceptual notation or mathematical logic do suggest such an interpretation. One such description is the following advertisement, which Peano published in Rivista di Matematica in 1892: “Of the greatest usefulness would be the publication of collections of all the theorems now known that refer to given branches of the mathematical sciences, so that the scholar would only have to look into the collection in order to find out what had been done concerning a certain point, and whether his own research was new or not. Such a collection, which would be long and difficult in ordinary language, is made noticeably easier by using the notation of mathematical logic; and the collection of the theorems on a given subject becomes perhaps less lengthy than its bibliography” (quoted in Kennedy 1980, p. 44). Peano even thought – and applied this thought to practice with little success and much hostility from his students – that the Formulario would be a useful instrument in the teaching of mathematics (Kennedy 1980, pp. 66, 100-101). There appears to be something, then, to Frege’s surmise that Peano’s intention was “orientated towards the storage of knowledge” and “towards brev- ity and international intelligibility” (1897, p. 237). Frege, of course, intended this as a criticism, too. He argued that Peano’s ideography was in fact unsuitable for the analytical and demonstrative purposes to which he had devised his own concept-script (id., pp.238-9). He suggested, furthermore, that such purposes may in fact not have been what Peano had in mind. See the next footnote. 13 That this kind of clarification was Peano’s intention is indicated, for example, by the title of his best-known work, Arithmetices principia, nova meth- odo exposita, which translates “The principles of arithmetic, presented by a new method”. Taking the title seriously, we ought to infer that Peano was concerned with the foundations as much (although not necessarily in the same sense) as any of those authors the titles of whose works feature such words as “Grundlagen” or “Grundgesetze” or “Principles”. The Preface to Arithmetices principia provides less circumstantial evidence: “Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. The difficulty has its main source 130 Ch. 3 Russell on Kant ography is both a language in which the concepts and propositions of mathematics can be represented in a perspicuous manner and a calculus in which proofs can be carried out. The presentation of a piece of mathematics in the language of Peano, in other words, is not simply an exercise in the translation from one language to another; rather, an ideography or mathematical characteristic, such as Peano’s, is grounded in an analysis of the concepts and reasonings found in real- life mathematics, which are then reproduced in the ideography in a manner that professes to be direct or transparent.14 Peano’s project, which Russell on one occasion described as the “reduction” of the in the ambiguity of language. That is why it is of the utmost importance to examine attentively the very words we use. My goal has been to undertake this examination, and in this paper I am presenting the results of my study, as well as some applications to arithmetic. I have denoted by signs all ideas that occur in the principles of arithmetic, so that every proposition is stated only by means of these signs. [...] With these notations, every proposition assumes the form and precision that equations have in algebra; from the propositions thus written other propositions are deduced, and in fact by procedures that are similar to those used in solving equations. This is the main point of the whole paper” (Peano 1889, p. 85). There is, then, a direct conflict between Peano’s appraisal of his own ideography and that which Frege gave (see the previous footnote). The proper resolution of this conflict is to attribute it to two sources: (i) Peano’s and Frege’s different conception of what is involved in “foundations”; (ii) certain details of Peano’s new method. 14 At one time Peano himself put the point as follows: “The transformation into symbols of propositions and proofs expressed in the ordinary form is often an easy thing. It is a very easy thing when treating of the more accurate authors, who have already analysed their ideas. It is enough to substitute, in the works of these authors, for the words of ordinary language, their equivalent symbols. Other authors present greater difficulty. For them one must completely analyse their ideas and then translate into symbols” (‘Con- cept of Number’, 1891; quoted in Kennedy 1980, p. 44). And in a more concise form: “[T]he notations of logic are not just a shorthand way of writing mathematical propositions; they are a powerful tool for analysing propositions and theories” (from the first edition of the Formulaire de Mathématiques; quoted in Kennedy 1980, p. 49). Ch. 3 Russell on Kant 131 whole of mathematics “to strict symbolic form” (Russell 1901a, p. 368), is therefore predicated on assumptions which could be seen to be of great foundational and philosophical importance. This is how Russell perceived the situation. Accordingly, he argued both in retrospect and immediately after the discovery of Peano’s mathematical logic that it constituted an instrument which made possible an exact treatment of the foundations of mathematics by extending “the region of mathematical precision backwards towards regions which had been given over to philosophical vagueness (1944, p. 12).15 In a manuscript entitled Recent Italian Work on the Foundations of Mathematics, which is his earliest semi-popular exposition of the foundations of mathematics, Russell explains how this extension takes place and why it is of such a tremendous importance for the philosophy of mathematics.16 The paper begins with a succinct sketch of the relevant mathematical context, which is followed by a description of Peano’s “new method”:

It is a well-known fact that mathematicians, under the influence of Weierstrass, have shown in modern times a care for accuracy, and an aversion to slipshod reasoning, such as had not been known among them previously since the time of the Greeks. The great inventions of the seventeenth century – analytical geometry and infinitesimal calculus – were so fruitful in new results that mathematicians had neither time nor inclination to examine their foundations. [...] Thus mathematicians were only awakened from their dogmatic slumbers when Weierstrass and his followers proved that many of their most cherished propositions are in general false, and hold only under exceptional circumstances. This rude shock led to an investigation, by mathematicians, of the correctness of the commonly accepted foundations of arithmetic and analysis: a movement analogous to that which the non-Euclideans produced in Ge- ometry. Dedekind, Cantor, and the other Germans, who have taken part in this work, are well known. But in this country, at any rate, neither

15 For further discussion, see section 1.5. 16 The paper was written for Mind. For some reason which remains unknown, however, it was not published there or anywhere else; see the edi- tor’s introduction to Russell (1901b). 132 Ch. 3 Russell on Kant

mathematicians nor philosophers, so far as I know, are aware that the finest work of all (as regards rigour of demonstration, not as regards the discovery of new theorems) has been done by a school of Italian philosophical mathematicians. [...] Its aim is, to discover the necessary and sufficient premisses of the various branches of mathematics, and to deduce results (mostly known already) by a rigid formalism which leaves no opening for the sinister influence of obviousness. Thus the interest of the work lies (1) in the discovery of the premisses and (2) in the absolute correctness of deduction. (1901b, p. 352)

At least in Russell’s view, the Peanists’ work derives its importance primarily from its concern with deductive rigour (“rigour of demonstration” or “absolute correctness of deduction”, as Russell calls it). The content of this notion will be unpacked in the following sections.

3.1.4 The Concept of Deductive Rigour

3.1.4.1 General Remarks

The notion of deductive rigour has both a local and a global dimension to it. Considered locally, it leads to a characterisation of rigorous reasoning; this amounts to explaining the notion of gap-free argument. Consid- ered globally, it leads to a characterisation of what is involved in a systematisation of all reasonings pertaining to a given branch of mathematics. In the ideal case, a description of systematisation is underwritten by an explicit local characterisation of rigour, or of gap-free argument. An alternative local characterisation could be given by using the notion of “elementarily valid inferential step”. For example, Philip Kitcher (1981, p. 469) describes rigorous reasoning as one which proceeds by means of elementary steps. This terminology may, indeed, be preferable in certain respects to the use of the notion of Ch. 3 Russell on Kant 133

“gap-free argument”.17 At any rate, we must distinguish between the following two senses of gap-free argument (and, therefore, between two senses of what it is to characterise an argument as gap-free): (i) an argument is gap-free relative to a system of inference if, and only if, there are no gaps in the argument according to the standards recognised by the system;18 (ii) an argument is gap-free in the absolute sense (i.e., consists of elementarily valid inferential steps) if, and only if, there are no gaps in the argument according to some standards which make no reference to any particular system of inference or set of accepted reasonings. Normally, claims to the effect that an argument or a set of reasonings is, or is not, rigorous, are normative claims: introducing rigour to an inferential practice is intended to improve that practice. Clearly, such claims have interesting and non-trivial content only if there is available an absolute notion of rigour, i.e., one that can be used, either by itself or by drawing on a particular system of inference that is taken to codify the absolute notion, in the evaluation of the reasonings licensed by this or that system or to compare two or more such systems for their rigour.19

17 Mainly because rigorization is seldom, if ever, a matter of “filling in the gaps”, as is, perhaps, suggested by the term “gap-free argument”. 18 “System of inference” is intended to cover not only such strict (rigorous) systems as can be found in logic-books – say, a system of natural deduction or the axiomatisations found in Begriffsschrift, Principia Mathematica, or Hilbert and Ackermann’s Principles of Theoretical Logic – but also less strictly codified sets of accepted reasonings, such as those that are in use in Euclid’s Elements. In the former case the gap-free nature of an “argument” is simply built into its definition. 19 Whether the availability of an absolute and normative notion of deductive rigour implies a commitment to absolutely elementary (“absolutely rigorous”) inferential steps is a further issue; on this point, cf. Kitcher (1981, pp. 481-482). 134 Ch. 3 Russell on Kant

For a mathematician, systematisation amounts to axiomatisation.20 The demand for global deductive rigour thus leads to a search for a (rigorous) axiomatisation of a given branch of mathematics. This goal is achieved when the mathematician has succeeded in identifying a set of primitive terms and primitive propositions (Russell’s “premisses”) which, relative to the branch under study, “contains, as it were in a nutshell, its whole contents”, as Frege (1885, p. 113) puts it; hence Russell’s requirement that the premises of a branch of mathematics must be “necessary and sufficient”.21 For a particular axiomatisation to be rigorous, everything that is recognised to belong to the field under consideration must have a place either among the primitives or else must be capable of being traced back to them; the second class, then, consists of terms definable by means of the primitive terms and propositions inferable (with the help of definitions, if needed) from the primitive propositions. Whether something is inferable from something else is of course determined by the particular notion of rigorous argument underlying the axiomatisation. That is, it is determined by whether the transition from the premisses to the conclusion

20 This statement oversimplifies, for axiomatisation is not the only form of mathematical systematisation. For example, Kitcher (1983, ch. 9.VI) distinguishes two main types of mathematical systematisation: axiomatisation and conceptualisation. His depiction of the former need not be repeated here; conceptualisation “consists in modifying mathematical language so as to reveal the similarities among results previously viewed as diverse or to show the common character of certain methods of reasoning” (p. 218). Clearly enough, the two types may occur together. In the current context, however, our interest is solely in axiomatisation. 21 The necessity-condition amounts to the requirement that axioms or premises should be independent. This feature was extensively studied in the Italian school (as Russell remarks in his (1901b)). The sufficiency-condition is to be understood in the sense of “experimental completeness”; in the paper under discussion Russell writes that the proof that the primitive propositions are sufficient is a “mere question of careful deduction” (id., p. 359). In the Principles, §120, the point is formulated as follows: a given branch of mathematics begins with “certain axioms or primitive propositions, from which all ordinary results are shown to follow”. Ch. 3 Russell on Kant 135 is gap-free in light of that notion. And then there is the further question, compulsory if a given axiomatisation is meant as a contribution to rigorization, as to whether the reasonings licensed by the axiomatisation capture an acceptable notion of “elementarily valid inference”, or at least a reasonable approximation thereof.

3.1.4.2 Pasch on Rigorous Reasoning

The demand for global deductive rigour or rigorous axiomatisation was gaining currency among mathematicians of the late nineteenth century, who were much occupied with the intricacies of the axiomatic-deductive method. There were significant differences among these mathematicians – and philosophers who were alert to mathematical developments – over the precise import of axiomatisation, and the notion of axiomatic-deductive method was presented under different guises. The Italian mathematicians were not the first to require that mathematical theories should be organised in the axiomatic-deductive form. Some earlier mathematicians had been occupied with systematisation and deductive rigour. Their need was first felt in abstract algebra and geometry, a development which Russell mentions in passing in the passage quoted above.22 The most notable of these geometers was Moritz Pasch, who treated the issue of deductive rigour at length in his Vorlesungen über neuere Geometrie, a work which purported to turn projective geometry into a “really deductive” science (Pasch 1882, p. 98). Pasch’s monograph is remarkable not so much for the programme of deductive rigour per se, for such programmes had been announced before him. What makes it important is the fact that it offers a workable notion of rigorous proof; as Freu- denthal (1962) observed, it was Pasch who taught mathematicians how to formulate their axioms. The following passage from Vorlesun- gen sums up Pasch’s notion of rigorous proof:

22 For the nineteenth-century developments in algebra, see Cavaillès (1962); for geometry, see Nagel (1939), Freudenthal (1962). 136 Ch. 3 Russell on Kant

In fact, if geometry is to be really deductive, the process of inference must everywhere be independent of the meaning of geometrical concepts, just as it must be independent of the diagrams; only the relations between geometrical concepts, as they have been specified in the statements and definitions employed, may legitimately be taken into account. During the deduction it may well be useful and legitimate, but in no way necessary, to think of the meaning of the relevant geometrical concepts: in fact, if it is necessary to do so, this makes manifest the incompleteness of the deduction and – should it not be possible to remove the gap by changing the reasoning – the insufficiency of the statements established previously for use in proofs. If, however, a theorem is rigorously deduced from a set of statements – we shall call them ‘root-statements’ – the derivation has a value which goes beyond its original purpose. For if correct statements can be obtained from the root-statements by replacing the geometrical concepts connected in them by certain other concepts, then the corresponding replacement in the theorem is admissible; without repeating the deduction we obtain in this way a (usually) new theorem, which is a consequence of the modified root-statements. (ibid.; all emphases in the original)23

23 The translation is taken from Nagel (1939, p. 237) with slight modifications. The original German reads: “Es muss in der That, wenn anders die Geometrie wirklich deductiv sein soll, der Process des Folgerns überall unabhängig sein vom Sinn der geometrischen Begriffe, wie er auch unabhängig sein muss von den Figuren; nur die in den benutzten Sätzen, beziehungsweise Definitionen niedergelegten Beziehungen zwischen den geometrischen Begriffen dürfen in Betrach kommen. Während der Deduction ist es zwar statthaft und nützlich, aber keineswegs nöthig, an die Bedeutung der auftretenden geometrischen Begriffe zu denken; so dass geradezu, wenn dies nöthig wird, daraus die Lückenhaftigkeit der Deduction und (wenn sich die Lücke nicht durch Abänderung des Raisonnements beseitigen lasst) die Unzulänglichkeit der als Beweismittel vorangeschickten Sätze hervorgeht. Hat man aber ein Theorem aus einer Gruppe von Sätzen - wir wollen sie “Stammsätze” nennen - in voller Streng deducirt, so besitz die Herleitung einen über der ursprünglichen Zweck hinausgehenden Werth. Denn wenn aus den Stammsätzen dadurch, dass man die darin verknüpften geometrischen Begriffe mit gewissen anderen vertauscht, wieder richtige Sätze hervorgehen, so ist in dem Theorem die entsprechende Vertauschung zulässig; man erhält so, ohne die Deduction zu wiederholen, einen (im Allgemeinen) neuen Satz, eine Folgerung aus der veränderten Stammsätzen.” Ch. 3 Russell on Kant 137

Like any other rigorization, Pasch’s axiomatic treatment of projective geometry purports to exhibit, or reconstruct, the relevant reasonings in a gap-free manner, that is to say, as series of elementarily valid inferential steps. Pasch’s conception of deductive rigour turns on the rejection of what may be called “verbal reasoning”. I borrow this apt term from Russell, who on one occasion used in to illustrate the differences between rigorous and non-rigorous mathematical reasoning (Russell 1901b p. 352). This term does not refer only to such reasonings as are formulated in a natural language, or a suitable extension thereof, even though such reasonings do fall under its scope. The distinctive feature of verbal reasoning is rather that it relies, either explicitly or implicitly, on the reasoner’s grasp of the meaning or content of some of the concepts featuring in the reasoning. The ban which Pasch imposes on meaning covers Euclid-style definitions of geometrical primitives: “a point is what has no parts”; “a line is length without breadth”; “a straight line is that which lies evenly with the points on itself,” are useless as definitions, which is shown by the fact that Euclid does not employ them in the actual derivation of theorems (Pasch 1882, pp. 15-16). More importantly, the ban also extends to whatever other meanings (and, hence, denotations) one might associate with one’s geometric vocabulary. It is true that such expressions as “point”, “line” and “plane” (or, rather, their German equivalents) feature in Pasch’s axioms, which are formulated in ordinary (mathematicians’) German, and he held the view – soon to become obsolete among working geometers – that the primitives possess meaning and denotation which can be understood only by reference to appropriate “objects of nature” (Pasch 1882, pp. 16-17).24 He wrote, furthermore, that axioms are statements with content that can be grasped only by inspecting corresponding geometrical figures (id., p. 43). The semantics of geometry, however, must not play any role in the derivation of consequences; whatever content and denotation the geometrical rea-

24 This change is usually associated with Hilbert (1899), but it is clearly observable in the works of Peano and his associates. 138 Ch. 3 Russell on Kant soner might wish to attach to his primitives, no reference to them is allowed in the course of a proof. What this ban on meaning means in practice can be seen by considering a sample set of axioms from Vorlesungen. The following five axioms, Pasch tells us, express the simplest observations concerning line segments and their points (Pasch 1882, §1, entitled “On the Straight Line)”:

1) Between two points it is always possible to draw one and only one line segment. 2) It is always possible to specify a point that is contained within a given line segment; 3) If the point C is contained within the segment AB, then the point A lies outside the segment BC; 4) If the point C lies within the segment AB, then all points of the segment AC are also points of the segment AB; 5) If the point C is contained with the segment AB, then a point which does not lie either within the segment AC or the segment BC, cannot lie within the segment AB (or, if the point C lies within the segment AB and the point D outside the segment AC, then the point D is contained within the segment BC).

From these axioms we can derive the following theorem:

If the point C is contained within the segment AB, and if the point D is contained within the segment BC, then the point C is contained within the segment AD.

The proof of this theorem is as follows:

Because the point D is assumed to lie within the segment BC, C lies outside the segment BD (by axiom (1)); because C lies within the segment AB, and D within the segment BC, D also lies outside the segment AB (by axiom (4)); because C and D Ch. 3 Russell on Kant 139

lie within the segment AB, and C lies outside the segment BD, C lies within the segment AD (by axiom (5)).

Figure 3.1 An illustration of a theorem from Pasch’s Vorlesungen

The example is exceedingly simple. It is precisely this simplicity, however, that brings out very clearly the central content of Pasch’s notion of deductive rigour. We can easily picture to ourselves a situation which verifies the antecedent of the conditional theorem; such a situation is depicted by Figure 3.1. And we can see that the situation we have imagined is one that verifies the consequent as well. To use a familiar word, the correctness of the theorem is obvious as soon as we have before our eyes a picture of a relevant situation. A further function can also be attached to such pictures: they are necessary for grasping the meaning of such terms as “point”, “line segment” and “contained within”. Consequently, we can say, as Pasch did, that geometric figures are useful both for comprehending the (intended) meanings of geometric terms and understanding geometric proofs. The activity of drawing pictures and reflecting on them is nevertheless not germane to doing geometry. For as Pasch insists, theorems are not justified by reflection on meanings and denotations, but must be proved exclusively from explicitly stated premises – axioms, definitions or statements which have already been proved (ibid.) This is what was done in the derivation above.25

25 This condition is one that Euclid fails to fulfil; his proofs often depend for their validity on the intended meanings of the geometric primitives, which are made evident by corresponding figures (Pasch 1882, pp. 43-44). Pasch illustrates this point by the very first proposition of Elements, the con- 140 Ch. 3 Russell on Kant

That meaning is irrelevant for inference in a properly axiomatized theory is shown, in particular, by the fact that correct deductions have a multiplicity of applications; if the axioms remain valid (or correct or true; richtig) under a replacement of one set of primitives by another, we obtain a corresponding set of theorems for the new set: this is what happens, for example, with the duality principle in projective geometry. Pasch’s approach to axiomatization and its underlying conception of rigorous proof proved to be seminal both within geometry and more widely. As regards the former, Pasch’s influence, evident in Hilbert’s Grundlagen der Geometrie,26 is described by Nagel as follows: “[t]he traits of logical purity and rigor which his [sc. Pasch’s] book illustrated became the standard for all subsequent essays in geometry. No work thereafter held the attention of students of the subject which did not begin with a careful enumeration of undefined or primitive terms and unproved or primitive statements; and which did not satisfy the condition that all further terms be defined; and all further statements proved, solely by means of this primitive base. In- deed, if anything, later writers have set for themselves standards of an even greater rigor of formalization than Pasch’s work exhibit” (1939, p. 239). Pasch’s influence is manifest also in Peano’s case. What is new about Peano’s new method, compared to Pasch, is, firstly, the extension of the demand for global rigour to cover the whole of mathematics. Secondly, Peano did precisely what Nagel says some mathematicians did after Pasch, to wit, he set mathematics even higher standards of rigour than Pasch’s work exhibit. On this point, there is a direct link between Pasch and Peano, for Pasch is one of those “more accurate authors, who have already analysed their ideas” and in whose works it is therefore “enough to substitute [...] for the words of ordinary language, their equivalent symbols,” as Peano puts it in struction of an equilateral triangle on a line segment as base (id., pp 44-5). Russell uses this same example in (1901a, p. 378) 26 See Contro (1976). Ch. 3 Russell on Kant 141 his “Sul concetto di numero” (“On the concept of Number”) of 1891.27 In doing so, he articulates in more detail than Pasch had done the relevant notion of rigorous reasoning. We saw above that Pasch’s conception of proper axiomatization amounts to drawing an absolutely sharp distinction between what could be called, respectively, the topic-sensitive content of a mathematical theory and its deductive component. In a theory that is adequately axiomatized, the topic-sensitive content is assigned to axioms and is thereby rendered maximally explicit.28 This is the point behind Pasch’s insistence that meaning is ir-

27 On Pasch’s influence on Peano, see Kennedy (1972) and Contro (1976). Once again, Nagel provides a succinct formulation: “In Italy, Pasch’s treatise became known through Peano’s translation of the substance of the Neuere Geometrie into the intuitively opaque notation of mathematical logic” (1939, pp. 239-240). Nagel does not tell which of Peano’s works he has in mind, but the reference could be to I principii di geometria logicamente esposti of 1889, in which Pasch’s work is discussed. 28 The proper treatment of the axioms and undefined terms is a question which allows a number of different answers. I shall return to these questions below, but the following preliminary observation may be in order. What Russell in the above quotation calls “premisses” have been dubbed “axioms”, “definitions”, “postulates”, “hypotheses” or simply “basic” or “primitive propositions” by other authors. The terminological choice may be insignificant, but often, at least when the views of a historical figure are being discussed, it is not, because it may indicate something important about the semantic or epistemic status which the author assigns to these ultimate premises and, more generally, to mathematical theories. If we imagine these different views arranged on a spectrum, we would find at one end the traditional conception of axiomatic theories, which is sometimes referred to as “material axiomatics”. In its refined form, this approach attributes such properties as unrelativized truth and/or self-evidence to the primitives with which the axiomatic development of a theory begins, but it is more self- conscious than the tradition about the conditions which a proper axiomatization should fulfil; this, roughly, was Pasch’s, and, in a different manner, Frege’s view. At the other end we would find that approach which com- mences with the so-called free postulation of axioms (definitions, etc.), from which the rest of the theory is developed purely deductively. This generic 142 Ch. 3 Russell on Kant relevant to geometrical inference. It does not mean that an axiomatized theory is meaningless or without content. Such a claim would be entirely at odds with Pasch’s philosophy of geometry. It means that topic-sensitive content in its entirety must be captured by axioms, and therefore that recourse to this content must not figure in the drawing of consequences. In another sense, however, reference to meaning continues to feature in the inferences that Pasch draws from his explicit premises; only this meaning now belongs to those items in the vocabulary that express the deductive component. In Pasch’s axiomatization the deductive component remains tacit – a practice not at all rare among working mathematicians, a famous case of this attitude being Hilbert (1899) – and hence the derivation of theorems is in fact founded on the reasoner’s grasp of the meanings that he assigns to certain of the words that he uses. In other words, consequences are “seen” to follow from their premises in somewhat the manner that Euclid “saw” his theorems to follow from what he found by the inspection of figures.29 If one follows Pasch in thinking that explicitness is the hallmark of deductive rigour, then the deductive component of a mathematical theory should be subjected to a treatment analogous to that given to its topic-sensitive content. This task is now assigned to logic.30 It constitutes a separate theory which can be studied independently of its use in actual reasoning. Its application to mathematical reasoning, furthermore, is to be made explicit: proofs, that is to say, are to be written out in full rather than merely indicated.31 approach (“deductivism”; cf. Resnik 1980, Ch. 3) admits a number of further refinements. 29 Cf. here Stenius (1978b). 30 The main reason why the systematic study of the deductive component should be called “logic” (and not something else) is presumably the old idea that logical vocabulary can be applied across disciplinary boundaries; cf. Shapiro (1997, p. 149). 31 The actual proofs that Peano present are logically speaking less than perspicuous; a fact that is plausibly attributed to Peano’s less than absolutely clear grasp on the distinction between a logical calculus and its metatheory Ch. 3 Russell on Kant 143

3.1.4.3 The Logicization of Mathematical Proof

The claim, implicit in Pasch and explicit in Peano, is thus that an absolute correctness of deduction can be attained only by logicizing mathematical proof.32 This goal, Peano thought, is best secured by formalization, i.e., by a “rigid formalism”, as Russell puts it, or again, by “banishing all words from our deductions, and effecting everything in a wholly symbolic language” (Russell 1901b, p. 352). Elsewhere Rus- sell writes that the modifier “symbolic” in “symbolic logic” marks out a merely accidental characteristic of the discipline that is no more than a “theoretically irrelevant convenience” (1903a, §11); the point is that logic is not about symbols any more than, say, geography or physics. This point is obviously an important one, and I shall return to it when I discuss Russell’s conception of logic in more detail.33 The fact remains, nevertheless, that the programme of deductive rigour is one that ultimately calls for formalization: for in this way it becomes possible sharply to delineate the deductive component of a theory, which is thus made amenable to a rigorous treatment. Peano’s new method thus consists in the representation of mathematical arguments in a special notation featuring two kinds of signs: (i) signs that are peculiar to this or that branch of mathematics; (ii) logical signs, or signs which

(Grattan-Guinness 1977, pp. 112-113; Zaitsev 1994). It is also true that his work on the deductive component (“logical calculus”) remained seriously incomplete. These two facts obscure the point made in the text, but they do not, I think, invalidate it. Peano, we should add, was not only an early cham- pion of rigour. He was also one of the first mathematicians (or logicians) to study the logical properties of mathematical theories. 32 Unsurprisingly, Peano saw that this move implied a crucial broadening of the scope of logic: “It is known that scholastic logic is of little use in mathematical proofs, seeing that in [them] the classification and rules of the syllogism are never mentioned, while on the other hand, one uses arguments that are entirely convincing, but not reducible to the forms considered in traditional logic” (from I principii di Geometria logicamente esposti, as quoted in Kennedy 1980, p. 28). 33 Cf. section 4.5.3. 144 Ch. 3 Russell on Kant alone are operative in mathematical reasoning. Deductions are now given a sure footing by transforming them into calculations which are effected by dint of rules that operate exclusively on the logical signs.34 This, in Russell’s opinion, is Peano’s great achievement, although the basic idea of transforming inference into calculation was of course first formulated by Leibniz:

To Peano [...] is due the revival, or at least the realization, of Leibniz’s great idea, that, if symbolic logic does really contain the essence of deductive reasoning, then all correct deduction must be capable of exhibition as a calculation by its rules. (1901b, p. 353)35

So far I have outlined the idea, present in Peano and appropriated by Russell, that deductive rigour is to be achieved by logicization and formalization. To recapitulate, the basic idea is to formulate mathematical proofs in a gap-free manner, i.e., in such a way that each and every inferential step is simple or elementarily valid. And this is now agreed to mean: all inferences must be purely logical in character. The aim of this exercise is the reconstruction and systematisation of mathematical reasonings – ultimately, entire mathematical theories – in a manner that renders them maximally explicit: once a reasoning is dissected into a series of steps that are simple or elementarily valid, every element that its conclusion depends upon can be identified and labelled appropriately.

34 Again, this does not mean that logical rules or rules of deduction are about symbols. It means only that a notation in which the line between logical and non-logical can be drawn sharply serves to make the deductive component explicit. 35 To be sure, this statement exaggerates the extent of Peano’s achievement. Clearly, not even a successful completion of Peano’s entire project would show that all correct deduction is logical in character: there are, after all, deductions outside mathematics, too. Nevertheless, Leibniz’s great idea does underlies Peano’s project, or at least Russell’s appropriation of it, so that it is natural to consider Peano’s ideography a partial realization of that idea. Ch. 3 Russell on Kant 145

3.1.5 Russell on Rigorous Reasoning

3.1.5.1 Self-evidence and Rigour

The chief characteristic which Russell attributes to non-rigorous mathematical practices is their reliance on obviousness or self-evidence. It is this characteristic that he singles out as the chief enemy of inferential correctness. He has two reasons for doing so. The first reason is the familiar phenomenon which 19th century mathematicians confronted repeatedly, to wit, that what appears to be self-evident or obvious may not even be true (or is true only when suitably restricted).36 The second reason, somewhat less obvious than the first but at least as significant, is that obviousness stands in the way of a proper articulation of mathematical reasoning; in other words, that obviousness obscures mathematical understanding. As we saw in chapter 1, the logicist Russell was mainly interested in the nature of what he called pure mathematics. This discipline is exclusively concerned with deduction or the drawing of consequences: “What we want to find out is, what can be deduced from what” is Russell’s shortest and most succinct description of the pure mathematician’s task (1901a, p. 367). Obviously, this task cannot be successfully completed as long as obviousness is regarded as the crucial attribute of mathematical propositions and inferences. Russell’s question – What can be deduced from what? – is likely to look less important if a proposition has a ring of obviousness to it (proving what is obvious is an activity derided already by the ancients); and

36 Standard examples, also used by Russell, include Weierstrass’ work in analysis and Cantor’s set theory. For example, generations of mathematicians and – in particular – philosophers were convinced of the truth of what Rus- sell calls the axiom of finitude (see, e.g., Russell 1903a, §188). Cantor, however, showed that this axiom holds only under special circumstances. Such examples show, furthermore, that not only is obviousness not an infallible guide to truth; there is the further possibility that a proposition that is initially highly counterintuitive may recommend itself for acceptance, once it’s content is made clear and put in a proper context. 146 Ch. 3 Russell on Kant should we manage to persuade ourselves that the question is legitimate, we are not likely to return a convincing answer to it as long as we have nothing but their alleged obviousness with which to back up our inferences. It is mainly for this reason, that is, to counter the influence of obviousness, that Russell recommends that a transition be made to Peano’s symbolism. He suggests, in effect, that it is only in this way that the real purpose of foundational studies can be secured:

It is not easy for the lay mind to realise the importance of symbolism in discussing the foundations of mathematics, and the explanation may perhaps seem strangely paradoxical. The fact is that symbolism is useful because it makes things difficult. (This is not true of the advanced parts of mathematics, but only of the beginnings.) What we wish to know is, what can be deduced from what. Now in the beginning everything is self-evident; and it is very hard to see whether one self-evident proposition follows from another or not. Obviousness is always the enemy to correctness. Hence we invent some new and difficult symbolism, in which nothing seems obvious. Then we set up certain rules for operating on the symbols and the whole thing becomes mechanical. In this way we find out what must be taken as premise and what can be demonstrated or defined. For instance, the whole of Arithmetic and Algebra has been shown to require three indefinable notions and five indemonstrable propositions. But without a symbolism it would have been very hard to find this out. (1901a, p. 368)37

In spite of his harsh judgment on obviousness, Russell is willing to leave some room and function for that notion in mathematical reasoning. To recognise the legitimacy of obviousness and, at the same time, to insist that, in foundational research, proofs are not to be judged by their obviousness or lack thereof, is to draw a distinction between two roles that may be assigned to inference (argument, proof, demonstration) in mathematics. On the one hand, there is the epistemic task of warranting mathematical belief. On the other hand,

37 Frege’s reasons for introducing his concept-script were broadly similar to Russell’s; see Frege (1884, §91). Ch. 3 Russell on Kant 147 there is the task of improving mathematical understanding. Russell, in other words, does not deny that obviousness, when suitably understood, is a property that can be attributed to at least some inferences and propositions of mathematics; indeed, he was at least on occasion willing to admit that obviousness may very well work as a perfectly reasonable criterion for the acceptability of a proposition or inference (1901b, pp. 357-8). To this, however, he makes the caveat that obviousness should be discarded as irrelevant when the task is not to convince someone of the correctness of a proposition but to discover and establish deductive relationships between propositions.

3.1.5.2 Different Sources of Self-evidence

In drawing this distinction Russell associates himself with an old tradition. For want of a better term, I shall call this tradition “foundationalism”.38 It goes back at least to Aristotle, who distinguished between proofs that something is thus-and-so and proofs which show why something is thus and so.39 In the 19th century, Bolzano made Aristotle’s distinction the cornerstone of his own foundational studies. Bolzano was uncommonly clear about the utility of proving the obvious: such proofs are not needed for epistemic purposes; they are needed to uncover the objective grounds of a theorem in focus.40 Similarly, Frege argued that proofs are not there – at least not primarily – to confer certainty on that which is proved; at least as important is the insight that a proof, once it is formulated in a rigorous manner,

38 Not to be confused with epistemic foundationalism. 39 Cf. here Stenius (1978a). 40 The best example of Bolzano’s idea is his famous paper on the Inter- mediate Value Theorem (Bolzano 1817). See also the Preface to Bolzano (1810). For Bolzano’s conception of foundations and the relevant notion of proof, see Kitcher (1976), Sebestik (1992, Ch. 1.IV and Ch. 2.IV). 148 Ch. 3 Russell on Kant permits into the interrelations holding between mathematical propositions (1884, §§4, 90).41 Russell, too, is very clear on this point. For example, he readily admits that most of the theorems which Peano derives using his formula language are “self-evident” (Russell 1901b, p. 357). This in no way diminishes the value of Peano’s work, however. To deduce correctly something from something else, Russell points out, is not merely a matter of “bringing out true results” (ibid.) Rather, correctness consists in “assigning a number of logical relations and rules of inference from which the rest can be deduced, in effecting this deduction in a manner free from fallacies [...], and in using no definable term until it has been defined” (id., pp. 357-358). It is the successful realisation of these requirements that make Peano, according to Rus- sell, “[t]he great master of the art of formal reasoning, among the men of our own day” (1901a, p. 368). The complaint that Russell makes here, i.e., that too much emphasis has been put on mathematical results instead of the process of reasoning itself is repeated a few years later in a passage which is Russell’s clearest espousal of foundationalism:

The true interest of a demonstration is not, as traditional modes of exposition suggest, concentrated wholly in the result; where this does occur, it

41 But why should there be a connection between the improvement of mathematical understanding and the establishment of deductive relationships? The answer to this question is that, in the present context, reasoning means more than just the drawing of consequences from a set of premises; it involves, also and crucially, an analysis of the content of the concepts featuring in the relevant reasonings; it is only when we have an adequate grasp of these concepts (say, the concepts of arithmetic) that we can say we really understand the reasonings delineated by these concepts (say, arithmetical reasoning). To borrow a useful pair of terms from Hintikka (1995), the foundationalist is interested not only in the deductive function of logic but also its descriptive function. Hintikka’s example of the descriptive function is the epsilon-delta definitions of concepts like continuity and differentiation (id., p. 9), which is of course a paradigm example of successful analysis for someone like Russell. Ch. 3 Russell on Kant 149

must be viewed as a defect, to be remedied, if possible, by so generalising the steps of the proof that each becomes important in and for itself. An argument which serves only to prove a conclusion is like a story subor- dinated to some moral which it is meant to teach: for aesthetic perfection no part of the whole should be merely a means. (1907a, p. 72)

It may seem that in one respect Russell’s appraisal of non-rigorous mathematicians is fundamentally unfair. For surely one hesitates to argue that the mathematicians of the past had nothing else to rely on in their inferences but the fact that they were “obvious”. Even if we admit that the reasonings by non-rigorous mathematicians were grounded on something more robust than mere mathematical in- stinct, we could nevertheless argue that obviousness or self-evidence continues to play a substantial role in mathematical reasoning. Here the following formulation suggests itself. An inferential step is acceptable as valid or legitimate (that is to say, reasoning is under proper control) when the reasoner finds himself in a position to judge that inferential transitions featuring concepts A, B, C,… are self- evident or obviously correct, where the judgment is informed by some salient features which the reasoner attaches to the relevant concepts. This formulation brings us back to the notion of verbal reasoning, which I mentioned in passing in connection with Pasch’s conception of deductive rigour. Thus, to give yet another formulation of the point just made, the real issue between Russell and non-rigorous mathematical reasoner concerns the source or ground of the judgments of self-evidence or obviousness.42

42 The adequacy of this formulation is conditional upon the assumption that Russell did admit some substantial role to the notion of self-evidence, even though it does not figure prominently in his conception of what the foundations of mathematics should be able to accomplish. I have already indicated that this is so. 150 Ch. 3 Russell on Kant

3.1.5.3 Logical Self-Evidence

For Russell, the legitimate ground of mathematical reasoning, i.e., the class of concepts which provide the admissible ground for assessing the validity or otherwise of an inference, is of course found in logic. Hence, insofar as there is a role for self-evidence to play in reasoning (and I have aleady indicated that this is so, according to Russell), the source of the judgments of self-evidence is the reasoner’s grasp of logical concepts. The non-rigorous reasoner’s attitude is conspicuously different. For him, the source or ground in question – the features of concepts that underlie legitimate inferential transitions – may remain more or less unarticulated, as something that is regulated only by an existing mathematical practice.43 The source can, however, be articulated in detail, either as a contribution to mathematical investigation (in which case the articulation amounts to rigorization44) or else as a contribution to a more philosophical project concerning the nature of mathematics (in which case the articulation amounts to an explanation why an existing practice in fact qualifies as a rigorous

43 Kitcher (1983, ch. 9.V) offers a useful reminder that working mathematicians need a special reason before they start to worry about rigour. Thus, for example, when Frege (1884) criticized mathematicians for being satisfied in their reasonings with “practical certainty”, which is grounded in a mass of successful applications, their attitude turns out to be perfectly reasonable (“rational”, to use Kitcher’s term), once it is seen that the attitude of working mathematicians towards rigour is dictated by how well existing conceptual tools permit them to handle actual research problems. What from the foundationalist standpoint may look like an illegitimate attitude (non- rigorous reasoning reigning supreme) may be, and often is, a perfectly rational attitude for a working mathematician to adopt. 44 There is an obvious complication here: As Kitcher (1981, p. 484; 1983, pp. 215-6) points out, rigorization may be either explicative or corrective. In the former case, rigorization is taken to show (explain, etc.) why a certain mathematical practice was successful. Often, however, rigorization may force a mathematician to force some reasonings that the older practice deemed successful. Thus, rigorization may or may not lead to a radical revision of a mathematical practice. Ch. 3 Russell on Kant 151 one; no doubt these two kinds of articulation proceed along very different lines, but in the present context this matters little). Either way, such an articulation constitutes a characterization of ‘rigorous reasoning’ that is, at least potentially, essentially different from the logical conception. For Russell, here lies the real problem with self-evidence; reliance on self-evidence or obviousness invites confusion as to the real nature of mathematical reasoning, because it invites “misplaced rigorization”. I shall return to this issue in the next section (3.2). Be- fore that, however, it is useful to provide further elucidation of the content of Russell’s foundationalism by elaborating on the distinction between the two functions of proof, epistemic and foundationalist. A fruitful way of doing this is by relating Russell’s views on mathematical reasoning to those held by Henri Poincaré.

3.1.5.4 Poincaré on Intuition and Self-evidence

Henri Poincaré never accepted the logicization of mathematical reasoning, and he criticized the proponents of what he called “new mathematics” (including Peano, Russell, Hilbert and Couturat) for distorting genuine mathematics and the spirit that alone made it alive and worthwhile (see Detlefsen 1992 and 1994 for details).45 According to Poincaré, the feature that separates mathematics from logic is the autonomy of mathematical reasoning. He argues, among other things, that one must separate genuine proofs from an activity that he calls verification (1902, pp. 3-4). An example of the latter is provided by Leibniz’s proof of the proposition that 2+2=4. It can-

45 In doing so he occasionally plunged into philosophy, where he defended Kant against the likes of Russell and Couturat: “in M. Couturat’s opinion, the new works, and more particularly those of Mr. Russell and Si- gnor Peano, have definitely settled the controversy so long in dispute between Leibnitz and Kant. They have shown that there is no such thing as an a priori synthetic judgment [...]; they have shown that mathematics is entirely reducible to logic, and that intuition plays no part in it whatever. [...] Can we subscribe to this decisive condemnation? I do not think so.” (1908, p. 146) 152 Ch. 3 Russell on Kant not be denied, Poincaré admits, that Leibniz’s proof is “purely analytic”. But precisely for this reason it is not a genuine proof at all but a mere verification in which the conclusion is nothing but the premises translated into a different language. The paradigmatic example of genuine proof is mathematical induction. Applied to natural numbers, this may be stated as follows:

P(0) For every n, if P(n), then P(n+1) For every n, P(n)

According to Poincaré, the essential characteristic of mathematical induction or “reasoning by recurrence” is that it condenses an infinity of analytical inferential steps into a single rule (1902, p. 9). He explains this as follows. The inferential steps contained in mathematical induction can be rewritten as follows:

P(0) If P(0), then P(1) P(1) If P(1), then P(2) P(2) If P(2), then P(3) P(3) . . .

Poincaré describes mathematical induction by saying that in it analytical inferences are arranged “in a cascade” (ibid.): the conclusion of each application of modus ponens serves as a premise for the next inferential step. These “particular consequences”, however, are not enough to establish the result that P holds for every natural number; to reach that conclusion, the reasoner would have to complete an infinity of inferential steps; hence, by Poincaré’s lights, he “should Ch. 3 Russell on Kant 153 have to cross an abyss which the patience of the analyst, restricted to the resources of formal logic, will never succeed in crossing” (id., pp. 10-11). Mathematical induction is therefore a genuinely mathematical mode of inference, or inference that cannot be reduced to purely logical reasoning. The essence of mathematical induction consists in the fact that it enables the mathematical reasoner to step from the finite to the infinite: “[t]his instrument [sc. mathematical induction] is always useful, for it enables us to leap over as many stages as we wish; it frees us from the necessity of long, tedious, and monotonous veri- fications which would rapidly become impracticable” (id., p. 11).46 Where, then, is the ground of validity for mathematical induction, according to Poincaré? Not amenable to analytical proof (or experimental verification or conventional grounding), mathematical induction is in fact an apriori and synthetic intuition. It is a capacity of the mind which “knows it can conceive of the indefinite repetition of the same act, when the act is once possible. The mind has a direct intuition of this power, and experience can only be for it an opportunity of using it and thereby becoming conscious of it” (id., p. 13). Poincaré’s conception of mathematical reasoning lends itself readily to a comparison with the foundationalist view. What he writes about mathematical reasoning is amenable to several different, more or less sophisticated, interpretations, but it does not seem altogether arbitrary to read his remarks on mathematical induction in the light of the epistemic conception of proof. That is to say, for Poincaré the

46 On the face of it, Poincaré’s conception of mathematical induction involves more than just one criticism of the “analytic view” (alternatively, and no less likely, he is just not scrupulous about the formulation). On the one hand, he seems to argue that formal logic is in principle insufficient for mathematical purposes: analytical proofs deal with particularities whereas genuinely mathematical reasoning proceeds from the particular to the general. On the other hand, he seems to suggest that mathematical induction is just “useful” in that it provides a mathematical reasoner with an inferential shortcut, as it were; clearly, the latter view is of little interest and should, in all probability, not be seen as central to Poincaré’s argument. 154 Ch. 3 Russell on Kant problem of mathematical induction is primarily that of justification or warrant, the question being: where do we find the legitimating source of induction, once logic and experiment and convention are excluded from fulfilling this role? What is perhaps most striking about Poincaré’s conception of mathematical induction is how casually he articulates it. As Russell points out in his review of Science and Hypothesis, mere indefinite repetition does not yet yield mathematical induction, i.e., it does not suffice to prove something about all natural numbers (Russell 1905b, pp. 71-72). Conceivably, we could say that mathematical induction is “grounded” in the possibility of adding 1 to a number, but, if we are to use induction in proofs about all natural numbers, we must add the further requirement that all natural numbers can be reached by a repeated application of this operation starting from 0 (or 1). Of course, Poincaré does recognize this when he writes that mathematical induction contains, condensed in a single formula, an infinite number of syllogisms. But here, surely, the question arises: how does he know this? What is needed, it seems, is not just a capacity of the mind, but an insight into the structure of natural numbers (Russell formulates this point by saying that the applicability of mathematical induction is a part of the definition of “natural number” (id., p. 72).47

47 Possibly, one could defend Poincaré by distinguishing between a capacity, that of indefinite iteration, and mathematical induction properly so- called; given this distinction, it could be said that the latter “presupposes” or “is grounded in” the former, but the two are nevertheless distinct. In that case, however, there remains the task of explaining how mathematical induction is validated. There is an evident connection between Poincaré’s notion of reasoning by recurrence and Kant’s notion of indefinite iterability as the source of arithmetic proofs and judgments (as Friedman (1996) points out). There is nevertheless also a conspicuous difference between the two. As we saw in section 2.4, Kant’s account of arithmetic does not leave much room for general truths about the domain of natural numbers; Poincaré, by contrast, is explicit that generality is a sine qua non for mathematics; “mathematicians”, he writes, “always endeavour to generalise the propositions they have obtained” (1902, p. 14). Ch. 3 Russell on Kant 155

As I read him, Poincaré simply takes mathematical induction for granted; for him it is a truth that “is imposed upon us with [...] an irresistible weight of evidence” (1902, p. 13).48 For the foundationalist like Russell, by contrast, the problem posed by mathematical induction is primarily that of making sense of its content, a problem that has to do with improving our mathematical understanding. It is precisely for this reason that the influence of obviousness – that which imposes itself upon us with an irresistible force – is so sinister, according to Russell: it obscures the very point of foundationalism as he understands it. According to Poincaré, the reason why mathematical induction is useful is that “it enables us to leap over as many stages as we wish”. For the foundationalist, this attitude obscures the very point of foundational research (in addition to the fact that such leaps are potential sources of error); accepting a conclusion of an inference or an inferential step itself on the grounds that it is obvious (as Poincaré seems to do) is to avoid the real question: “What exactly is involved in this kind of reasoning, say, mathematical induction?” Once this question is taken up, logicization is not the only answer available; rigour is not inevitably a matter of formal logic, as there are other answers available, too.49 For a foundationalist like Russell, however, alternative analyses of rigour were simply consequences of what

48 According to some scholars, there is more to Poincaré’s conception of mathematical induction than this. On a not implausible construal, his answer to the question as to why mathematical induction is valid is basically Kantian or at least quasi-Kantian in character: mathematical induction is justified – or, at any rate, we are justified in using it; the line between the two is not clear – because “reasoning by recurrence” is a necessary feature of the mind, something that is “necessarily imposed on us” (Poincaré 1905, p. 13), and, as such, is presupposed in experience. (See Folina (1992, Ch. 2) and (1996) for attempts to give coherent content to Poincaré’s notion of the synthetic apriori.) The point remains, nevertheless, that Poincaré’s actual explanation of the connection between mathematical induction and our mental capacities is deficient in the way indicated. 49 See Detlefsen (1992) and (1994) for a Poincaréan development of this theme. 156 Ch. 3 Russell on Kant was referred to as “misplaced rigorization” above. This charge figures prominently in Russell’s criticism of Kant.

3.2. Kant and Misplaced Rigorization

3.2.1 Russell and Kant on Mathematical Reasoning: the Standard Reading

We are now in a position to examine the details of Russell’s criticism of Kant’s theory of mathematics. Here the focus is on Kant’s conception of mathematical reasoning. What Russell has to say about Kant is well- known, at least in outline. This criticism, however, is often formulated in a manner which does not do justice to the real content and scope of Russell’s views. Standard discussions of Russell’s criticism presuppose the following charting of what Kant might have meant by the thesis that mathematics is synthetic and apriori. This thesis, it is said, could mean either of the following:

1) that some of the axioms (or definitions) of a mathematical theory are synthetic (“intuitive”); 2) that some inference steps in mathematical proofs are synthetic (involve the use of “intuitions”).

Given these two options, it is customarily thought that in Russell’s view Kant subscribed to 2). In this way the received view focuses, and sees Russell as focussing, on what may be called mathematical reasoning in the narrow sense, or the derivation of consequences from their premises.50 The received view continues by attributing to Russell the following diagnosis of why Kant should have thought the way he did. Ac-

50 Examples of the standard interpretation are Beck (1955), Hintikka (1967), (1969a), Brittan (1978, p. 42) and Shin (1997). Ch. 3 Russell on Kant 157 cording to this diagnosis, Kant’s conception of mathematical reasoning was a natural outcome of the following two assumptions; I shall formulate these with reference to Kant’s theory of geometry, as it is the standard context of discussion:

A. Kant’s conception of formal logic. According to Kant, formal logic had reached finality and was incapable of improvement. In particular, the different modes that together constitute syllogistic reasoning are the only formally valid types of reasoning.

B. Euclid’s presentation of geometry. This provides a striking example of a type of deductive reasoning which falls outside the scope of formal logic.51

According to this diagnosis, Kant was led to think, through reflection on existing mathematical proof practices, that there is a non- accidental distinction between logical and non-logical kinds of deductive reasoning. Proofs like those one finds in Euclid’s Elements contain inferential transitions which are not licensed by the rules of formal logic. To explain why mathematical reasoning is nevertheless correctly felt to be a species of deductive reasoning, Kant introduced the notion of construction in pure intuition. Russell’s diagnosis is of course followed by criticism. Briefly, his argument is that the development of formal logic and its application to mathematical theories has rendered Kant’s conception of mathematical reasoning outdated; “intuition” can be dispensed with in

51 Here I use the term “deductive reasoning” as a general term for any kind of reasoning that is necessarily truth-preserving. Kant, of course, knew very well that not all reasoning is “logical” (ordinary induction, for example, is not), but the present point is that, according to him, there are types of inferences where the conclusion is a necessary consequence of the premises but where this necessity of consequence cannot be captured by the rules of formal logic. 158 Ch. 3 Russell on Kant geometrical proofs simply by reformulating them as sequences of purely logical inference steps.52 The received view is of course not incorrect. There are passages which clearly show that Russell did attribute to Kant the view that mathematical proof procedures are essentially extra-logical. The best- known and most elaborate of these is found in the Principles, §4:

There was, until very lately, a special difficulty in the principles of mathematics. It seemed plain that mathematics consists of deductions, and yet the orthodox accounts of deduction were largely or wholly inapplicable to existing mathematics. Not only the Aristotelian syllogistic theory, but also the modern doctrines of Symbolic Logic, were either theoretically inadequate to mathematical reasoning, or at any rate required such artificial forms of statement that they could not be practically applied. In this fact lay the strength of the Kantian view, which asserted that mathematical reasoning is not strictly formal, but always uses intuitions, i.e. the à priori knowledge of space and time. Thanks to the progress of Symbolic Logic, especially as treated by Professor Peano, this part of the Kantian philosophy is now capable of a final and irrevocable refutation.

This passage also contains an explanation of the origin of Kant’s theory. The point is elaborated in §434:

[F]ormal logic was, in Kant’s day, in a very much more backward state than at present. It was still possible to hold, as Kant did, that no great advance had been made since Aristotle, and none, therefore, was likely to occur in the future. The syllogism still remained the one type of formally correct reasoning; and the syllogism was certainly inadequate for mathematics. [...] [I]n Kant’s day, mathematics itself was, logically, very inferior to what it is now. It is perfectly true, for example, that anyone who attempts, without the use of the figure, to deduce Euclid’s seventh proposition from Euclid’s axioms, will find the task impossible; and there probably did not exist, in the eighteenth century, any single logically correct piece of mathematical reasoning, that is to say, any reasoning which

52 On this view, the elimination of “intuition” from geometrical proofs seems to amount to no more than the elimination of diagrams. Ch. 3 Russell on Kant 159

correctly deduced its result from the explicit premisses laid down by the author. Since the correctness of the result seemed indubitable, it was natural to suppose that mathematical proof was something different from logical proof. But the fact is, that the whole difference lay in the fact that mathematical proofs were simply unsound (Russell 1903a, §434).

This latter passage is useful in that it forges a connection between what Russell says in this passage and what was suggested in the previous section regarding self-evidence or obviousness. The point Rus- sell is making in §434 is simply this. Finding that a proposition is indubitable (self-evident, obvious) discourages one from searching for a proper proof of a proposition; even worse is the possibility that the correctness of the conclusion lends unwarranted credibility to existing proofs and positively encourages one to accept such proofs as rigorous, i.e., to think that they consist of elementarily valid inference steps. Kant accepted this view and devised a corresponding explanation as to why this should be so even if such proofs fell outside the purview of formal logic as he knew it.53

53 Some twenty years earlier, Frege formulated the same diagnosis in very clear terms (although Kant is not mentioned here by name): “A single such step [sc. inferential step which is self-evident] is often really a whole com- pendium equivalent to several simple inferences [...]. In proofs as we know them, progress is by jumps, which is why the variety of types of inference in mathematics appears to be so excessively rich, for the bigger jump, the more diverse are the combination it can represent of simple inferences with axioms derived from intuition. Often, nevertheless, the correctness of such a transition is immediately self-evident to us, without our ever becoming conscious of the subordinate steps condensed within; whereupon, since it does not obviously conform to any of the recognized types of logical inference, we are prepared to accept its self-evidence forthwith as intuitive” (1884, §90). 160 Ch. 3 Russell on Kant

3.2.2 Some Remarks on the Standard View

The standard view, though correct as far as it goes, is not the whole truth about Russell’s attitude towards Kant’s theory of mathematics. Two points need to be made here. The received view, we saw above, is articulated against the background of a contrast between two interpretations of the role that intuitions play in Kant’s theory of mathematics. According to the first interpretation, intuitions are there to provide a suitable grounding for the axioms or definitions of a mathematical theory (and “grounding”, it seems, means simply truth). The second interpretation suggests that intuitions are needed, roughly, to mediate mathematical inferences. We could call the first interpretation externalist, as it considers the primary role of intuition to lie outside the axiomatic system itself. The second interpretation, an internalist one, argues for a more intimate connection between intuition and mathematical concepts. On this view, mathematical concepts must be related to sensibility – must be constructible in pure intuition – because this gives them whatever content they have. And this giving of content (mathematical content, as it might be called) is quite independent of the further issue of the empirical applicability of mathematical concepts.54 There are more than one reason why one might want to insist on the intuitive character of the axioms (or definitions) of a mathematical theory. Here I shall mention only one such reason; I will return to the question towards the end of this section. Some Kant-scholars argue, as Gordon Brittan has done, that “[t]he synthetic character of the propositions of mathematics is a function of some feature of the propositions themselves and not of the way in which they come to be established” (Brittan 1978, pp. 55- 6). In defending this interpretation, they are likely to have the problem of applicability in mind. That is, they may argue that Kant insisted on the constructibility of all mathematical concepts, not because construction are needed to account for mathematical reasoning, but be-

54 Here I am of course recapitulating points made in Chapter 2. Ch. 3 Russell on Kant 161 cause he saw in this a way to secure the application of mathematics to the physical sciences and, therefore, to the real world. This idea naturally suggests a very simple and straightforward defence of Kant against Russell. If Kant’s reasons for thinking that mathematics is synthetic were not the reasons identified by Russell, then it can be argued that Russell’s criticism simply misses its intended target. Or, at any rate, it can be argued that it ignores those questions that were uppermost in Kant’s mind, when he developed his account of mathematics. Cassirer (1907, sec. VI) gives a forceful formulation of this idea. He argues that, for all its success in the analysis of mathematical ideas, “logistics” – that is, the new mathematical logic of Russell et al. – fails to address the critical question, first asked by Kant, concerning the possibility and presuppositions of objective, empirical knowledge:

That, however, presents a problem which is completely outside the horizon of “logistics” and which is therefore not touched by its criticisms. All empirical judgments are beyond its scope: it stops on the border of experience. What it develops is a system of hypothetical suppositions, of which we can never know whether they are ever actualized in some experience, and whether, therefore, they ever allow of a mediate or immediate concrete application. (id., p. 43)55

According to Cassirer, it is only within the sphere of experiential science (“Erfahrungswissenschaft”), that the concepts of logic and mathematics receive their true legitimation (ibid.) The requirement that there should be a logic of objective knowledge (“eine Logik der gegen- ständlichen Erkenntnis”; id., p. 44) is something that lies outside the

55 My translation. The original German reads: “Damit ist aber ein Problem gestellt, das völlig ausserhalb des Gesichtskreises des “Logistik” liegt und das somit von ihrer Kritik auch nicht berührt wird. Alle empirische Urteile liegen jenseit ihres Bereiches: sie macht an der Grenze der Erfahrung Halt. Was sie entwickelt ist ein System hypothetischer Voraussetzungen, von denen wir aber niemals wissen können, ob sie sich jemals in irgend einer Erfahrung verwirklicht werden, ob sie daher jemals irgend eine mittelbare oder unmittelbare konkrete Anwendung verstatten werden.” 162 Ch. 3 Russell on Kant scope of logistics. As Cassirer points out it in the quotation above, Russell’s mathematical logic is a system of purely hypothetical judgments, and all empirical judgments are therefore outside its immediate scope. It is only when it is clearly understood that the very same syntheses which underwrite the formation of concepts in mathematics and logic also govern the building of experiential knowledge that these principles receive their true justification (id., p. 45). Hence, the criticisms that Russell and other proponents of the new logic level against Kant leave Kant’s real problem intact: this is the transcendental problem of the relation of mathematics to empirical objects.56 Cassirer’s appraisal of the conceptual situation, although it contains an important insight, fails to do justice to Russell’s “logistics”. There are two reasons for this failure. Firstly, the question concerning the role of mathematics and logic in experiential science is simply not on Russell’s agenda in the Principles, which is exclusively dedicated to the philosophy of pure mathematics. It is true that Kant’s critical problem is a real one, and Russell never denied this. But he is not to be criticized for failing to answer a question in a work that is not calculated to answer that question in the first place. He himself pointed this out, showing at the same time that he was well aware of the different functions that one might assign to intuitions:

But admitting the reasonings of Geometry to be purely formal, a Kantian may still maintain that an à priori intuition assures him that the definition of three-dimensional Euclidean space, alone among the definitions of possible spaces, is the definition of an existent, or at any rate of an entity having some relation to existents which other spaces do not have. This opinion is, strictly speaking irrelevant to the philosophy of mathematics, since mathematics is throughout indifferent to the question whether its entities exist. Kant thought that the actual reasoning of mathematics was different from that of logic; the suggested emendation drops this opinion, and maintains merely a new primitive proposition, to the effect that Euclidean space is that of the actual world. Thus, although I do not be-

56 See Richardson (1998, pp. 116-122; 128-129) for a more detailed description of Cassirer’s criticism of “logistics”. Ch. 3 Russell on Kant 163

lieve in any immediate intuition guaranteeing any such primitive proposition, I shall not undertake the refutation of this opinion. It is enough, for my purpose, to have shown that no such intuition is relevant to any strictly mathematical proposition (1903a, §434; emphases in the original).57

Russell’s task in the Principles is to develop a satisfactory philosophy of mathematics which accords with “modern mathematics”. That “mathematics” here means pure mathematics is not an arbitrary delineation of a subject-matter or a decision to use an old word in a new sense. On the contrary, in concentrating exclusively on pure mathematics Russell takes himself to be developing a philosophy that satisfactorily reflects the actual mathematical practice. To repeat, the pure mathematician is only interested in developing the deductive consequences of a set of axioms (definitions, postulates). The further question, whether, say, this or that geometry is true of the actual world – the problem of application – is simply irrelevant here. Thus, also, the problem setting of critical philosophy – the development of a logic of objectual knowledge – is irrelevant for Russell’s purposes. Secondly, Cassirer seems to think that Russell’s logistics implies a flatly empiricist stand on the critical question (Cassirer 1907, p. 43); logistics implies, he argues, that the task of thought (as opposed to sensibility) is limited to the sphere of pure mathematics. This in turn means that the potential empirical content of mathematical concepts is given entirely through uninformed sense-perception: even though his attitude

57 I do not wish to suggest that the possibility envisaged by Russell in this passage was Cassirer’s way of resolving the critical problem. The issues here are complex, and they include, among other things, a description of the different strategies that the neo-Kantians developed to circumvent the problems which the invention of non-Euclidean geometries created for critical philosophy. In particular, Cassirer would have rejected the simple strategy of arguing that even though non-Euclidean geometries are logical possibilities, Euclidean geometry alone is consistent with the facts of our “space- perception”; though common among neo-Kantians, this view is far too sim- plistic to suit Cassirer, who had thoroughly absorbed the methodological lessons taught by Poincaré’s conventionalism (see Richardson 1998, Ch. 5). 164 Ch. 3 Russell on Kant towards Kant was by no means uncritical, Cassirer, a self-avowed neo-Kantian, found this kind of flat empiricism thoroughly mistaken. This verdict on logistics is not correct, however. The theme propounded by critical philosophers, the identification of the principles underlying the construction of “objective knowledge”, is one that began to occupy Russell’s mind only a few years later, after the completion of Principia. The position which he then developed, though in complete agreement with anything he had said in the Principles, is far from any kind of flat empiricism. Let us now return to Russell’s criticism of Kant’s conception of mathematical proof. The second problem with the standard view is that if the externalist interpretation of intuition threatens to make that criticism irrelevant in a rather uninteresting way – by concentrating on the issue of empirical applicability, which was not germane to Russell’s logicism – the difficulty with the internalist reading, as it emerges in the context of the standard view, is that exclusive concern with mathematical reasoning in the narrow sense fails to highlight what is really at stake in the dispute between Russell and Kant. Re- verting to a piece of terminology introduced in chapter 2, it fails to highlight the representation-theoretic issues which lie at the heart of both Kant’s theory of mathematics and Russell’s foundationalism. These issues, it must be emphasized against the standard view, have to do not only with mathematical reasoning in the narrow sense (the deductive role or function of logic); equally important is the descriptive function of logic, i.e., the role that logic plays in the representation of the content of mathematical concepts. Indeed, it is not difficult to see that the former is dependent upon the latter; it is clearly impossible to “logicize” the deductive component of a theory without imposing a sufficient amount of logical structure on the concepts featuring in that theory; that is to say, it is impossible to logicize a piece of mathematical reasoning without analysing the ideas featuring in the relevant reasonings. The descriptive role, on the other hand, is not in this way dependent on an explicit formulation of the deductive component. For once the content of concepts is rigorously developed, the question as to what deductions can be effected from them can be and Ch. 3 Russell on Kant 165 often is answered “intuitively”, without the help of an explicitly formulated set of rules of inference.58 It is nevertheless hard to see how the descriptive function of logic could be worked out without observing the felt inferential relationships holding between concepts; the content of concepts, one is inclined to say, is articulated precisely by drawing on the inferences that one is permitted to make from given concepts. This is what happens in Kant’s theory of mathematics, too, except that for him the descriptive and deductive roles are not performed by formal logic but by intuitive constructions or intuitive representations: for mathematical purposes the content of a concept is revealed by studying the set of legitimate constructions that can be performed on it. Summing up the discussion so far, there are two things wrong with the standard view of Russell’s criticism of Kant. Firstly, the context in which the received view is usually formulated is strongly sug- gestive of an interpretation of Kant that seriously misrepresents the actual line of thought that Kant followed in his theory of mathematics. It does this by suggesting that Kant’s famous question, “How is pure mathematics possible?” is only about the applicability of mathematics. If this is accepted, it must be concluded that Russell’s criticism was grossly beside the point. However, on the interpretation defended in chapter 2 and attributed here to Russell, Kant was no less concerned with the conditions of mathematical thought per se. This leads to the second point, which is the claim that the received view is too narrow: mathematical proof or reasoning in the narrow sense does not exhaust the semantic or content-related problems which vexed Kant and Russell and which forms the gist of the latter’s criticism of the former.

3.3 Russell’s Criticisms of Kant

58 Pasch’s work on projective geometry is a good example of this attitude: even though he was the first to transform a substantial piece of mathematics into a really deductive science, the logic underlying Pasch’s axiomatization remained at an intuitive level. 166 Ch. 3 Russell on Kant

3.3.1 Russell on intuitions

Consideration of the representation-theoretic issues leads to a further issue, the role that logicism plays in Russell’s criticism of Kant’s theory of mathematics. Russell’s rejection of Kant’s conception of mathematical proofs – mathematical reasoning in the narrow sense – has nothing to do with logicism per se. This can be seen, for instance, by considering §4 of the Principles. There Russell maintains that the refutation of Kant’s conception of mathematical proofs is due to Peano’s symbolic logic (and, hence, not to his own logicism). In other words, mere rigorization or logicization of mathematical proofs suffices to show that Kant’s views on what is involved in mathematical reasoning in the narrow sense were thoroughly erroneous. If, therefore, we concentrate exclusively on the narrow sense, we are bound to give an incomplete picture of Russell’s criticism of Kant’s theory of mathematics; if, that is, we assume that logicism did play a role in that criticism. This assumption is one that will be vindicated in the sequel. Russell’s criticism of Kant’s theory of mathematics is put into proper focus by asking how Russell understood the term “intuition” and the role that Kant assigned to it in his theory of mathematics. What has been said so far already suggests that Russell’s understanding of the crucial Kantian term, though evidently not a result of detailed textual studies, is not as naive as is sometimes thought. Above all, Russell is perfectly correct, as against the externalist interpretation of “intuition”, when he sees the gist of Kant’s theory in his semantics of mathematical concepts (“semantics”, it was agreed in chapter 1, is not a term that Russell would have used at this time, but this is irrelevant). For Kant, the basic reason why intuitive conceptualizations (of a certain specified kind) are needed in mathematics is not that they forge a connection between abstract mathematics and the spatiotemporal reality;59 they are needed to make mathematical reasoning pos-

59 According to Longuenesse (1998, p. 290), “[t]he problem of the nature of mathematical thinking interests Kant only insofar as solving it might help Ch. 3 Russell on Kant 167 sible in the first place.60 And above we saw how Russell dismissed the application-problem as irrelevant to his own concerns; the implication of this is that it was irrelevant for Kant’s purposes as well; or at least the implication is that Kant’s account of mathematical reasoning can be considered having significance that is independent of the question of applicability. It is the semantic or representation- theoretical views that Russell has in mind when he argues that Kant’s notion of pure intuition is “wholly inapplicable to mathematics in its present form” (1901a, p. 379), i.e., that it cannot be used to elucidate the newly invented disciplines to which the term “pure mathematics” is applicable. It must be admitted that no extended discussion of “intuition” is to be found in the pages of Principles. What little Russell says by way of a direct characterization is found in sections 4 and 433. In the former section “intuition” is paraphrased as “the apriori knowledge of space and time”; in the latter section this is elucidated, if only slightly, by saying that, according to Kant, “the propositions of mathematics all deal with something subjective, which he calls a form clarify the conditions of possibility of experience and thus, more generally, the relation between cognitive subjects and objects of cognition and of thought.” What is said in the text does not contravene this statement. The claim made in the text is only that in Kant’s eventual solution of the problem of the relation of mathematical thought to the objects of experience, the semantic problem of the content of mathematical concepts comes first, and this solution then suggests the characteristically Kantian way of resolving the transcendental problem. 60 The logicist Russell, to be sure, was not much interested in the conditions of possibility of anything: for him this notion was a confused admix- ture of psychological and logical elements (see section 3.3.6.3) The conditions that make mathematical reasoning possible in the Kantian sense, however, are not merely – and perhaps not even primarily – conditions of grasping the content; they are as much conditions of the content itself. Even if vitiated by “psychologism” in the manner indicated by Russell, the Kantian notion does have a legitimate non-psychologistic core, which is constituted by the representation-theoretic or content-related issues which were dealt with in chapter 2. 168 Ch. 3 Russell on Kant of intuition. Of these forms there are two, space and time: time is the source of Arithmetic, space of Geometry.” Such statements do not point in the direction of any very precise notion of intuition: what they show is merely that Russell thought that, according to Kant, there is some connection between mathematics, on the one hand, and space and time as the two “forms of intuition”, on the other. Considered on their own, these passages say almost nothing about the role that Russell thought intuitions might play in Kant’s theory. And it is quite conceivable that he did not have anything very detailed or precise in mind when he commented upon Kant. It is possible, then, that Russell was content with a very general criticism which he based on the following two observations: (i) that there is an evident connection between Kant’s notion of pure intuition and certain mathematical practices which were permeated with conceptualisations making essential use of spatial and temporal notions; (ii) these practices are supplanted by modern mathematics, which is characterized more than anything else by a tendency to dispense with a not very precisely delineated set of intuitive conceptualizations, or conceptualizations which build on spatiotemporal notions. It must be admitted that this is a possible reading. A closer reading of the Principles allows us to conclude, however, that there was more to Russell’s understanding of Kant than the schematic argument just presented. This claim can be further supported by considering his earlier, pre-logicist work on the principles of mathematics. For before his encounter with Moore’s incipient realism and, more decisively, Peano’s mathematical logic, Russell’s numerous attempts to provide “an analysis of mathematical reasoning” were all of them carried out against a broadly Kantian background. There is no doubt that Russell’s later criticisms of Kant – such as we find in the Princi- ples, for instance – were not based on detailed exegesis. What makes Russell’s acquaintance with Kant more than casual and lends content to his appraisal of Kant’s theory of mathematical reasoning is the fact that he had himself experimented with views not too dissimilar from those expressed by his predecessor. It is therefore not implausible to suggest that when the logicist Russell wrote his comments on Kant’s Ch. 3 Russell on Kant 169 theory of mathematics, he had his own former self in mind as much as Kant; it is likely, therefore, that his understanding of Kant was substantially influenced by his own earlier work.

3.3.2 Russell’s Kantian Background

The pre-logicist Russell’s semantic Kantianism can be illustrated by how he proposed, at one time, to deal with the traditional problems surrounding the notions of quantity and number. What he has to say about these problems – more precisely, about the connection between our “number-concept” and “discrete quantity” – shows how little progress even reasonably well-informed philosophers had made since the publication of Kant’s first Critique. More to the point, it shows that Russell had absorbed a good deal of Kant’s theory of mathematics. Concerning what he calls the “number-concept”, the idealist Russell has this to say:

Number primarily derived from instances of a concept: purely intellectual from the start. Abstracts from concept of which they are instances, and pays attention merely to the iteration. We have, in number proper, a unity, but not a unit. Fractions, irrationals, imaginaries, etc. arise from introducing notion of a unit. We have strictly, in number, two unities, one a complex whole, containing several of the smaller unity. But the unity of the whole is very loose, in that it is merely formal: it is supposed to derive, from its being a whole, no quality but that of formal unity. The simpler unities are regarded apart from all qualitative differences, in fact, qua unities in number, they have no qualitative differences. But they are discrete, and the unity is prescribed, not arbitrary. (1896-98, p. 13)

Number, throughout the following discussion, will only be used of dis- creta; it will be taken as always the result, not of comparison as to the more or less, but of acts of synthesis (or analysis) of things whose qualitative or quantitative differences are disregarded. Pure number will denote the formal result of acts of synthesis, so far as any result can be known in total abstraction from the matter synthesized and from the 170 Ch. 3 Russell on Kant

specific qualities of the objects of the synthesis (1896b, p. 46; the passage is repeated in Russell. (1897b, p. 71)).

These two passages show sufficient similarity to Kant’s account of numerical judgments to warrant the claim that there was more to Russell’s grasp of Kant’s notion of intuition than is, perhaps, commonly admitted. The import of these passages and their precise connection with Kant’s ideas is admittedly not entirely transparent. I venture to offer the following reconstruction. The pre-logicist Russell held, as did many others at the time, that the fundamental application of numbers is to counting; the primary meaning of “number” is thus explained through an analysis of counting. Counting, in turn, was considered an activity whereby a multiplicity is measured by iterations of a single element, or unit determined by a common concept (here I follow the explanation given by Griffin 1991, p. 232). Hence, and here Russell comes close to Kant, the actual application of the number concept – “applied number”, as Russell (1896b, p. 47) calls it – presupposes “pure number”, a more abstract or “formal” notion which constitutes the “form” of any actual act of counting. This concept of pure number consists of two further notions: (i) unity (multiplicity considered as a single entity) and (ii) iteration (determination of the quantity of the multiplicity, which yields an answer to the question, “How many Xs are there?” or “What is the number of Xs?”, where X is the unit-determining common concept). These two notions seem to presuppose each other: a unity is something that is given – or generated or synthesized – by iteration, i.e., by a repeated application of one and the same act, so that the act itself is a unity of a kind. As Russell puts it, “[s]ince a unit must be defined by some quality, pure number will thus have no reference to a unit, or rather its unit is the abstract object of any act of attention, of whatever kind this may be” (id., p. 46). In the second of the passages quoted above Russell says that the most fundamental number concept, or the process generating it, is “purely intellectual”. Griffin (1991, p. 231, fn. 3) offers a plausible explanation of why he should have thought this. For Russell – as for Ch. 3 Russell on Kant 171

Bradley, who was a major influence on Russell at the time – discrete things or entities to which numbers in the fundamental sense are applied are not given in direct sense experience but presuppose conceptualization. Kant’s view is broadly similar; after all, according to his famous words, “[t]houghts without content are empty, intuitions without concepts are blind”, which in turns means that it is “just as necessary to make our concepts sensible, that is, add the object to them in intuition, as to make our intuitions intelligible, that is, to bring them under concepts” (B75). As this passage emphasises, the necessity of conceptualization did not lead Kant to say that number is something “purely intellectual”; rather, the notion of schema, which is the key to Kant’s theory of mathematics, has both a conceptual and an intuitive aspect to it.61 Whether the concept of number is purely intellectual is an issue that is in the Kantian tradition determined by one’s views on synthesis: the question is whether there can be a purely intellectual synthesis or whether, as Kant thought, synthesis results only from the interplay of concepts (understanding) and intuitions (sensibility). Russell’s position on this issue is probably not very clear. Like his idealist contemporaries, however, he thoroughly rejected Kant’s way of drawing the analytic-synthetic distinction (1897c, §56). This strongly suggests a substantial modification to Kant’s conception of the faculties and their interrelations, but it does not over- turn a genuinely Kantian notion of synthesis and, hence, a genuinely Kantian theory of judgments of number. These considerations suffice to show that Russell’s earliest attempts at analysing arithmetical reasoning were heavily influenced by Kant. These attempts, however, were very short-lived. In An Analysis of Mathematical Reasoning, a manuscript on which he worked from the April to the July of 1898, Russell sketched a rather different philosophy of arithmetic (for details, see Griffin 1991, Ch. 7.3). According to Griffin (id., p. 270), “the work marks the culmination of Russell’s attempts to fashion a Kantian philosophy of pure mathematics in which the concept of quantity played a central role”. Judging from

61 This was explained in more detail in section 2.3.1. 172 Ch. 3 Russell on Kant the substance of that work, the Kantian connection is in fact rather thin, a fact which Griffin, too, points out (ibid.).62 Nevertheless, the Introduction to that work still suggests a strong adherence to Kant. Russell argues there that most of the “conceptions” that belong to mathematics are intermediate between those that derive, via abstraction, from sense-data and those that belong to “pure logic”.63 Russell calls the latter conceptions “pure categories”, whereas the fundamental or indefinable mathematical conceptions are referred to as “categories of intuition” (Russell 1898a, pp. 164-165). Their characteristic mark is that they “express some aspect or property of space or time or both, or of whatever is in space or time” (id., p. 164). Unfortu- nately, we do not know how Russell thought this idea should be elaborated; those parts of the manuscript where the Kantian connection would have been most obvious – the topic of quantity and number – are either missing or were not written at all.

62 This should not be attributed to the circumstance that Russell’s Kant- ianism was spurious. Rather, it is due to the fact that Russell’s mathematical philosophy was rapidly changing at the time. If we assume that the short chapters of the Analysis were composed in the order in which they were arranged in the manuscript, we may conclude that the Russell of July 1898 was much less Kantian than the Russell of April 1898. An Analysis of Mathematical Reasoning is in fact the first work by Russell which shows the influence of Moore’s “new realism”; the details are given in Griffin (1991, Ch. 7.2). 63 By the notions belonging to “pure logic” is meant, roughly, such notions as belong to the general analysis of judgments. In the manuscript under consideration, these notions are dealt with in a chapter entitled “The Ele- ments of Judgments” (Russell 1898a, pp. 167-73), and they include at least the following: relation, unity, diversity, subject, predicate, term, logical subject, being, intension, extension, existence, quality, attribute and content. Ch. 3 Russell on Kant 173

3.3.3 Quantity in the Principles

As we have seen, the pre-logicist Russell developed the traditional quantity conception of mathematics from a broadly Kantian perspective. In the Principles, the mathematics underwriting this approach are firmly rejected in favour of arithmetization:

Among the traditional problems of mathematical philosophy, few are more important than the relation of quantity to number. Opinion as to this relation has undergone many revolutions. [...] The view prevailed that number and quantity were the objects of mathematical investigation, and that the two were so similar as not to require careful separation. Thus number was applied to quantities without any hesitation, and conversely, where existing numbers were found inadequate to measurement, new ones were created on the sole ground that every quantity must have a numerical measure. All this is now happily changed. [...] Weierstrass, Dedekind, Cantor, and their followers, have pointed out that, if irrational numbers are to be significantly employed as measures of quantitative fractions, they must be defined without reference to quantity; and the same men who showed the necessity of such a definition have supplied the want which they had created. In this way, during the last thirty or forty years, a new subject, which has added quite immeasurably to theoretical correctness, has been created, which may legitimately be called Arithmetic; for, starting with integers, it succeeds in defining whatever else it requires – rationals, limits, irrationals, continuity, and so on. It results that, for all Algebra and Analysis, it is unnecessary to assume any material beyond the integers, which, as we have seen, can themselves be defined in logical terms. It is this science, far more than non-Euclidean Geometry, that is really fatal to the Kantian theory of à priori intuitions as the basis of mathematics. Continuity and irrationals were formerly the strongholds of the school who may be called intuitionists, but these strongholds are theirs no longer. Arithmetic has grown so as to include all that can strictly be called pure in the traditional mathematics. (1903a, §149)

When the “number concept” and arithmetical reasoning are explained in the manner described above, to wit, by connecting it with discrete quantity (usually through some such process as counting), there arises 174 Ch. 3 Russell on Kant the further problem of accounting for continuous quantity. Without an adequate, purely arithmetical theory of reals, an explanation of the latter must be sought elsewhere:

It was formerly supposed – and herein lay the real strength of Kant’s mathematical philosophy – that continuity had an essential reference to space and time, and that the Calculus (as the word fluxion suggests) in some way presupposed motion or at least change. In this view, the philosophy of space and time preceded the Transcendental Dialectic, and the antinomies (at least the mathematical ones) were essentially spatiotemporal. All this has been changed by modern mathematics. What is called the arithmetization of mathematics has shown that all the problems presented, in this respect, by space and time, are already present in pure arithmetic. (id., §249)

Russell argues in these quotations that earlier mathematicians had committed the mistake of not paying enough attention to questions of meaning.64 An example is provided by Newton:

[...] Newton was, of course, entirely ignorant of the fact that his Lemmas depend upon the modern theory of continuity; moreover, the appeal to time and change, which appears in the word fluxion, and to space, which appears in the Lemmas, was wholly unnecessary, and served merely to hide the fact that no definition of continuity had been given. (id., §303; italics added)

The situation is changed only with the “growth of purely arithmetical mathematics”, as Russell calls it on one occasion (1899-1900, p. 106). Intuitive conceptualizations are now no longer needed for an under-

64As Nicholas Griffin pointed out to me, the claim that Russell is arguing that not enough attention has been paid to questions of meaning invites misinterpretation in view of Russell’s complaints in the Principles about meaning being a confused notion (for this issue, see below, section 4.4.1). As Griffin observed, though, Russell’s point is representation-theoretic, one that concerns content. Hence the complaint is entirely in keeping with Russell’s neglect of meaning, when meaning is understood in some other (“psychological”) sense. Ch. 3 Russell on Kant 175 standing of the concepts of the calculus; space and time become, at most, fields to which certain more abstract theories can be applied. Thus, Russell is in a position to argue that, in a discussion of continuity and infinity, “space and time need only be used as illustrations, not as vital to the principles involved” (ibid.) He also argues that the changes which have taken place in mathematics (the provision of proper meanings or definitions) is also highly relevant for the philosophy of mathematics: “[i]t is only in the last thirty or forty years that mathematicians have provided the requisite mathematical foundations for a philosophy of the Calculus; and these foundations, as is natural, are as yet little known among philosophers, except in France” (Russell 1903a, §303; the French are acquitted of the charge on account of Couturat’s doctoral dissertation, De l’Infini mathématique, which had appeared in 1896). It turns out, in particular, that the requisite mathematical foundations leave no room for the notion of intuition, developed in the manner of Kant: “[w]e shall find it possible to give a general definition of continuity, in which no appeal is made to the mass of unanalyzed prejudice which Kantians call ‘intuition’” (Russell 1903a, §249; italics added). Analogous observations are applicable to the more basic parts of arithmetic. For example, Russell dismisses what he on an earlier occasion calls “the addition-theory of number”(Russell 1898a, p. 213) in a passage which sounds almost like a comment on his earlier self, who had written that the unit of measurement associated with “pure number” was “the abstract object of any act of attention, of whatever kind this may be”: Some readers may suppose that a definition of what is meant by saying that two classes have the same number is wholly unnecessary. The way to find out, they may say, is to count both classes. It is such notions as this which have, until recently, prevented the exhibition of Arithmetic as a branch of Pure Logic. For the question immediately arises: What is meant by counting? To this question we usually get only some irrelevant psychological answer, as, that counting consists in successive acts of attention. In order to count 10, I suppose that ten acts of attention are required: certainly a most useful development of the number 10! Counting has, in fact, a good meaning, which is not psychological. But this meaning is 176 Ch. 3 Russell on Kant

highly complex; it is only applicable to classes which can be well-ordered, which are not known to be all classes; and it only gives the number of the class when that number is finite – a rare and exceptional case. We must not, therefore, bring in counting where the definition of number is in question. (1903a, §109; emphasis added)

This charge is repeated some sixty years later. The criticism contained in the following passage, it should be noted, does not do justice to the counting- or ratio-theory of number; the main point, however, remains unaffected:

The philosophy of mathematics was wrongly conceived by every writer before Frege. The mistake that all of them made was a very natural one. They thought of numbers as resulting from counting, and got into hope- less puzzles because things that are counted as one can equally well be counted as many. [...] It is obvious [...] that what makes anything one from the point of view of counting is not its physical constitution but the question, ‘Of what is this an instance?’ The number that you arrive at by counting is the number of some collection, and the collection has whatever number it does have before you count it. It is only qua many instances of something that the collection is many. The collection itself will be an instance of something else, and qua instance counts as one in enumeration. We are thus forced to face the question, ‘What is a collection?’ and ‘What is an instance?’ Neither of these is intelligible except by means of propositional functions. (1959, p. 68) Ch. 3 Russell on Kant 177

3.3.4 Propositional Functions in the Principles

Though deficient in details, this passage from My Philosophical Devel- opment is useful in that it focuses on the notion of propositional function. Originally due to Peano, this technical innovation enabled Russell to construct a viable philosophy of mathematics. In the passage quoted he argues, as against earlier philosophers of mathematics, including Kant and his former self, that the notion of number (cardinal number) can be correctly understood only with the help of the notion of propositional function: cardinal numbers are numbers of “collections”, and what collections are – the real meaning of collection, as this notion is used by modern mathematicians – can only be grasped by grasping the notion of propositional function.65 We should note, however, that the real reason why Russell came to think that propositional functions are indispensable was not the problem of “one and many” but problems connected with infinity.66 He spells out the real reason in the Principles, §338. His starting-point is the idea of a propositional function, that is, the idea of an entity which yields a proposition, when the variable or variables occurring in it are given constant values.67 The propositions thus formed are either true or false and the values for which the function is true form a collection or class. Thus, a class or collection is definable by a propositional function and the actual enumeration of the members of the class becomes irrelevant.

65 Whether Russell is correct in thinking that the notion of set that is relevant for” modern mathematics” is in this way tied up with propositional functions is a question which I shall bypass here. 66 For details, see Levine (1998, 96-98). 67 The details surrounding the notion of propositional function are of course extremely complex. Russell’s official line of thought in the Principles appears to be that for “technical development”, the notion may be accepted as a primitive. In §338 and elsewhere in the Principles he argues, however, that it is in fact analysable by means of the notions of “proposition” and “constituent of a proposition”. For some details, see sections 4.4.8 and 4.49 below. 178 Ch. 3 Russell on Kant

Russell’s argument here is directly related to Kant’s philosophy of mathematics (and, it may be added, to his own earlier writings). For Kant had had, and Kantians continued to have, difficulties in comprehending the actual infinite: if numbers and collections must be underwritten by enumeration or “synthesis”, all of them must be finite; at most, they can be potentially infinite.68 And Kant had a very good reason for maintaining that this necessity is genuine. For he held that there can be no purely analytic, or logical, or conceptual representation of the infinite; hence, this notion is a legitimate one only to the extent that it can be represented intuitively, by means of a “synthesis” that involves both concepts and intuitions. If, by this line of thought, we are lead to impose rather strict limitations upon the kind of infinity that is legitimate, the proper response is that such limitations are in no way arbitrary but follow from the very conditions of thought about the infinite. Russell’s position, after he had become under the influence of Peano, is crucially different in this respect. Kant’s difficulty is got rid of by means of the notion of a propositional function: this technical device puts Russell in a position to maintain that infinite classes are legitimate entities, even though no enumeration or “synthesis” could ever yield them: classes are defined

68 Thus, the difficulties that Kantians had in comprehending the actual infinite were usually formulated as reasons for resisting the postulation of such entities. For example, the Kantian Russell argued in his review of Cou- turat’s De l’Infini mathématique that “one would have supposed that the condition of being a completed whole, which he [sc. Couturat] has urged as necessary to number, would have precluded the possibility of infinite number” (Russell 1897a, p. 63). Russell is willing to admit the intelligibility of Coutu- rat’s contention that a collection is given as a whole as soon as its defining condition has been given, and he says that this is “the only hope of saving infinite number from contradiction” (ibid.) However, he asserts, as against Couturat, that any condition which could be so used involves or presupposes an “addition of elements” (id., pp. 65-6). From this follows the claim that “mathematical infinity consists essentially in the absence of totality” (ibid.). This profoundly Kantian idea leads to conclusions that directly contradict Couturat’s “infinitism”. Ch. 3 Russell on Kant 179 by propositional functions, and “the actual enumeration of the members of a class is not necessary for its definition” (ibid.) These considerations concerning the infinite illustrate a more general point: the notion of propositional function was Russell’s chief technical device which enabled him to overcome Kant’s “intuitive” semantics. In this respect it is analogous to Frege’s notion of function. Logic, after its scope is decisively extended by means of the notion of propositional function (Peano, Russell) or function (Frege), is capable of providing mathematics with a “grammar”, as Russell (1959, p. 66) calls it. That is to say, logic constitutes a general framework with the help of which mathematical ideas – mathematical reasoning in the broad sense – can be both analysed and represented. It is precisely this feature that first recommended Peano’s mathematical logic to Russell. Of course, the new logic of Frege, Peano and Russell is capable of overcoming Kantian semantics precisely because the two are, so to speak, functionally equivalent. That is, the new logic is powerful enough to reproduce those synthetic modes of reasoning which Kant had argued were beyond the scope of “pure general logic” and which he had assigned to a “special logic” (the logic of mathematical reasoning). As we have already seen, Kant held the view that the syntheticity of synthetic reasoning is a consequence of its being concerned with instantiation; this distinguishes special logics from general logic, in which predication is a relation between species- and genus-concepts.69 Synthetic reasoning, in other words, is capable of representing individuals qua individuals, and not merely qua schematic satisfiers of simple and complex predicates (in the limited sense of “predicate” recognized by general logic). It can be said, then, that Kant did introduce at least one significant, and genuinely logical innovation with his idea of “transcendental logic”;70 he argued, however obliquely, that the correct form of predication, as far as the logics-with-content go, is not the traditional “S is P”, a relation between terms or concepts,

69 See above, section 2.1.4.2. 70 For this point, see Thompson (1972, p. 96). 180 Ch. 3 Russell on Kant but “Fx”, a relation between an object and a concept.71 This innovation was incorporated into formal logic by Frege and other logical re- formists, when they criticized the traditional doctrine of subject and predicate and explained instantiation by means of the notion of (propositional) function. In this way the latter notion became the basis of a general analysis or decomposition of propositions or other such information-conveying entities.72

3.3.5 Against Russell: the Notion of Intuition Again

So far we have seen that Russell’s most basic charge against Kant was the contention that certain advances in mathematics – most importantly, the so-called arithmetization – had rendered Kant’s theory superfluous. That theory is grounded in an “intuitive semantics” for mathematical concepts, according to which such concepts are given

71 The early Russell was less explicit about concept-object distinction than Frege. This is largely due to the fact that his analysis of propositions was founded on the single category of a term (not to be confused with the terms of traditional logic!), i.e., a constituent of a proposition (I shall discuss this matter in detail chapter 4). In practice, however, Russell observed the distinction. For example he did so by recognizing the difference between class-membership and class-inclusion; for this distinction, see, for example Russell (1901b, pp. 354-7). 72 Cassirer (1907, p. 36) argues that if by synthesis in its most general sense we mean, as Kant did, “the act of adding different representations together, and of grasping their manifoldness in a single cognition” (A77/B103), this can be taken as a circumlocution for the methods that underlie the logical calculus, as this is conceived by Russell and Couturat. The import of the special “thought-devices” of logistics, like the part-whole relation, the general notion of function or the relations of identity and difference, is not captured by the principle of contradiction; rather, they are though-procedures for the “constructive creation of new content”. Although Russell would, in all probability, not have accepted Cassirer’s particular formulation of the point, he did agree with its substance; after all, he insisted on the synthetic character of the new logic. Ch. 3 Russell on Kant 181 content via a synthesis of the manifold of intuition, “through which we first give ourselves an object and generate its concept” (A234/B267). This proto-semantics explains, among other things, why the inferences performed by mathematicians are correctly regarded as rigorous: they are rigorous because of the special constitutive role that synthesis plays in the generation of mathematical content.73 Such intuitive conceptualizations are now replaced by purely logical developments of the relevant concepts and reasonings; in this way the representation-theoretic function of intuition is taken over by formal logic. We have already seen how one might try to vindicate Kant against “logistics”; one might argue that the real import of critical philosophy, as regards mathematics, lies in the problem of its empirical application, or its application to natural science; more precisely, once could admit, as Cassirer did, that Kant’s theory of mathematics must be subjected to several modifications in the light of the new logical and mathematical methods and nevertheless go on insisting that the application of these new methods to natural science must be resolved one way or another. But even if one sidesteps this problem of empirical applicability (for instance by insisting on the distinction between pure and applied mathematics), there remains a problem that is internal to axiomatized mathematical theories, i.e., to pure mathematics. For one could argue – and some did argue – that mere rigorous, logical axiomatization of a mathematical theory does not suffice to dispense with intuition; even after it is recognized that the new logic is capable of overthrowing intuition as the semantic ground of mathematical conceptualizations and reasonings, there remains the following problem of the indefinables: how are the axioms and primitive concepts featuring in those axioms to be understood? And here one could refer to something like intuition. According to some commentators, this was Kant’s own view; he is said to have endorsed the view that the syn-

73 More precisely, mathematical reasonings are rigorous, according to Kant, because their ultimate basis is to be found in the simplest operations of the mind, like the successive addition of homogeneous units. 182 Ch. 3 Russell on Kant theticity of mathematics does not lie in its method (inferences guided by intuition) but in the fact that its primitive concepts and axioms must be accessible to or grounded in intuition. This view is formulated in the following passage by Lewis White Beck:

The real dispute between Kant and his critics is not whether the theorems are analytic in the sense of being strictly deducible, and not whether they should be called analytic now when it is admitted that they are deducible from definitions, but whether there are any primitive propositions which are synthetic and intuitive. Kant is arguing that the axioms cannot be analytic [...] because they must establish a connection that can be exhibited in intuition. (1955, pp. 358-9)

Whether or not Kant held the view attributed to him by Beck is a contentious matter. It is not clear, either, whether the view he endorses in the quotation is one that concerns pure or applied mathematics; it is not clear, that is, whether the requirement that axioms must establish a connection between concepts that can be exhibited in intuition is meant to forge a connection between mathematical concepts and empirical objects or whether the question concerns the axioms of pure mathematics. It is at least possible, and perhaps even likely, that the former view is what Beck has in mind; nevertheless, the quotation can be read in the latter way, too. Be these questions as they may, Beck’s contention can be seen as congenial to a defender of Kant. If it can be sustained, it shows, pace Russell, that no amount of formalization and logical analysis is capable of dispensing with intuition; even after all analysis, formalization and rigorization has been completed, intuition must be invoked as the source of the primitive concepts and axioms of a theory. It is here that Russell’s logicism makes itself felt. For the different explanatory strategies that Russell subsumed under the label “logicism” all have one common goal: they purport to show that all of pure mathematics is exclusively concerned with logical notions.74 If

74 This requirement means one thing in arithmetic and something quite different in geometry. In arithmetic the primitives of Peano’s axiomatization Ch. 3 Russell on Kant 183 the strategies are successful, there is no point, as far as the deductions effected by the pure mathematician go, where something extra-logical needs to be introduced. In the absence of the techniques which together constitute Russell’s logicism, one would have to recognize, as Russell himself still did in the autumn of 1900, that “[w]hat distinguishes a special branch of mathematics is a certain collection of primitive or indefinable ideas, and a certain collection of primitive or indemonstrable propositions concerning these ideas” (1901b, p. 353). What logicism is intended to guarantee, then, is that nothing beyond logic is needed for the understanding of pure mathematics: that logic suffices for mathematical reasoning in the broad sense is shown by rigorization; that is suffices to eliminate the indefinables qua extra- logical entities is shown by logicism proper. It is natural to formulate the import of logicism in terms of the notion of intuition: if “logic” and “intuition” are given senses that exclude each other, then “rigor- ism” plus logicism together demonstrate that intuition is not needed for the understanding of pure mathematics. This characterization is bound to remain controversial, as long as there is vagueness in the sense which one gives to intuition. This point can be seen by considering the description that Gregory Landini has offered of the philosophical point behind Russell’s logicism. He starts by arguing – correctly – that Russell's point is about intuition. Then he goes on to write:

In Russell’s view, the appearance that there are uniquely mathematical intuitions is produced, from the one side, by logic’s having been underde- veloped, and from the other, by a lack of rigor in the construction of proofs in some branches of mathematics (e.g. geometry). The purpose of deduction of mathematical formulas within the formal calculus for logic is to demonstrate that intuitions which were thought to be uniquely mathematical are, in fact, logical.

are eliminated through explicit logical definitions, whereas geometric indefinables are got rid of by the twin-strategy of constructing them as variables and by conditionalizing geometric propositions (cf. section 4.6.6) 184 Ch. 3 Russell on Kant

Consider, for example, mathematical intuition. This seems to be grounded on a uniquely mathematical and non-logical intuition of the sequence of the numbers. Russell intends to show that the intuition is logical after all. This is to be accomplished by demonstrating that mathematical induction can be derived from logical principles alone. This does not deprive the mathematician of her intuitions – as if mathematical knowledge requires only knowledge of a few logical primitives. The claim is that the intuitions of mathematics are just those of logic and our knowledge of logic is synthetic a priori and rich with intuition.[...] There is a genuine intuition of the sequence of the natural numbers underlying mathematical intuition, but the logicist finds the intuition to be a purely logical intuition revealed by a logical analysis of the nature of number. (1998, p. 16).

Landini’s characterization of Russell’s early logicism is not very helpful, because he gives no indication as to how he thinks the crucial terms “intuition” and “logical intuition” should be understood. But if I may venture a guess, Landini’s interpretative idea is to forge a connection between intuition and syntheticity, i.e., he seems to attribute to Russell the view that since logic is synthetic (non-analytic), it is also intuitive; otherwise there would be no legitimate move from mathematics’s being synthetic apriori to its being “rich with intuition”. This, however, is not very helpful either, since syntheticity is not given a further explanation. For my part, I would put Russell’s point slightly differently, saying that the early Russell’s intention was to develop a new theory of the synthetic apriori, as this occurs in (pure) mathematics. Such a theory would dispense with Kantian intuitions and replace them with the new formal logic. The new logic is therefore synthetic in the clear and straightforward sense that it is not analytic in the extremely narrow sense which Kant had given to that term. This, however, does not imply that logic is intuitive in anything but a nominal sense. The dispute over whether Russell admitted intuition in some more robust sense – sense that is more than purely nominal – sense is bound to remain verbal as long as we do not fix upon one, sufficiently well delineated sense of “intuition”. This is precisely what I have tried to do in the preceding sections. I have argued that the pri- Ch. 3 Russell on Kant 185 mary sense which he gave to Kant’s “intuition” is apriori knowledge of space and time and that it is this sense which he purports to do away with his logicist philosophy of mathematics. There are, no doubt, other possible senses of intuition – for instance intuition qua singular representation – which are not undermined by the pursuit of rigour and logicism.75, 76 I shall, however, continue to endorse the view just explained, as it seems to me to be the clearest as well as closest to the relevant texts.77

75 For instance, in his interpretation of Kant, Hintikka equates intuitivity with individuality, thus separating its meaning from what the Transcendental Aesthetic adds to it. Hintikka’s interpretative strategy is correct to the extent that individuality is what Ansschauung means for Kant: Kantian “intuitions” are, by definition, singular representations (cf. section 2.1.4.3). Given this strategy, Hintikka is in a position to argue that the introduction of modern logic in no way undermines Kant’s theory. After all, singular reference is present in predicate logic as much as it was in Kant’s theory of constructions. Russell, by contrast, clearly thinks that spatio-temporality is an essential ingredient in Kant’s notion of intuition (cf. Russell 1903a, §249). On this particular point, I am inclined to agree with Russell, rather than Hintikka. Hintikka’s observation about the meaning of Anschauung is undeniably correct. It does not follow from this, however, that there is no essential or necessary connection between intuitions and space-time, for Kant; there is, in the sense that objects can only be given to us in space and time. 76 There are, also, other senses of “intuitive” which are undermined by logicism and the pursuit of rigour. In particular, there is the popular or col- loquial sense of intuitive as self-evident. As we have seen, one of the points behind the rigorization of mathematical proofs was to dispense with intuitive proofs in this sense; that is, the rigorization of proof was intended to substitute logical proofs from explicit premises for proofs grounded in self- evidence. This sense of intuition is not necessarily incompatible with the more specifically Kantian sense. 77 There is also the further point that in discussing the Russell-Kant connection one had better find a sense of the term “intuition” that can be fruitfully applied to both philosophers. In this sense the example given in the text, to wit, intuition qua singular representation, is less than helpful, in my opinion, since it is too thin to support the representation-theoretic theses that arguably constitute the core of Kant’s philosophy of mathematics. 186 Ch. 3 Russell on Kant

3.4. Summary

Russell’s new theory of the synthetic apriori plays a crucial role in his criticism of Kant’s theory of mathematics. But its importance goes far beyond that, for it constitutes the starting-point for a general anti- Kantian (or, one might argue, even anti-idealist) argument. Scholars sympathetic to Kant have argued that a Russellian argument from the nature of mathematics is at most of a limited importance. In arguing for this conclusion, they have argued in different ways that the new logical techniques, even the putative reduction of mathematics to logic, do not suffice to undermine Kant’s philosophy. For the questions with which Kant was concerned are still there, even though the framework in which they are now formulated must undergo a non- trivial change in the light of later developments. By the framework I mean the following three distinctions or contrasts:

- synthetic vs. analytic - sensibility vs. understanding - intuition vs. logic

According to Kant, in each case the items mentioned on the left-hand side coincide: the sphere of the synthetic is the sphere of the sensibly given, which is the sphere of (human) intuition. Similarly for the items on the right-hand side: formal logic issues analytic judgments, whose ground is found in understanding or pure thought (I ignore the complication that pure thought has for Kant an object-related use, which marks the transition from formal to transcendental logic). When, however, formal logic is given new content, this Kantian picture is bound to change, but, importantly, Kant’s questions do not thereby lose their vitality. Thus, for example, Cassirer pointed out, as against Louis Couturat, that the “purely logical nature of mathematical judgments” does not suffice to demonstrate that they are analytic:

It is the fundamental deficiency of Couturat’s criticism that he never keeps these two viewpoints apart: for him “synthetic” means concepts Ch. 3 Russell on Kant 187

and propositions which flow from the intuition, whereas “analytic” refers to those that can be grounded in pure thought. This way of determining the concepts, however, is an obvious petitio principii: in it the resolution of the problem is already anticipated in the formulation of the question. For without doubt critical philosophy, too, recognizes forms of pure intellectual synthesis, which must be related to the intuition of space and time in order to provide a foundation for empirical knowledge, but which nevertheless owe their truth and validity to “pure understanding”. (Cassirer 1907, pp. 35-36; emphasis in the original)78

A point similar to Cassirer’s is raised by Theodore De Laguna in his review of Couturat’s Les Principes des Mathematiques. Couturat’s work which provides an exposition of Russell’s philosophy of mathematics together with a sharply critical essay on Kant’s philosophy of mathematics (published earlier as Couturat 1904):

As regards the criticism of Kant contained in the appendix, a less favourable judgment must be given. The author’s logical machinery is fa- tally inadequate to his task. In detail, the criticism is irreproachably correct; but it leaves the fundamental issues where it found them. For, after all, the result of the whole argument is but to add another leading question to the Kantian Prolegomena: How is pure logic possible? The reduction of formal logic to a system of independent postulates simply throws into relief the fact that these postulates, at least, are synthetic propositions, which a Kant might well assume to be a priori, and into whose justification he would then certainly proceed to inquire. Perhaps, however, for this very reason, the essay should have an unusual interest for the appre-

78 My translation. The original German reads: “Es ist der Grundmangel von Couturats Kritik, dass sie diese beiden Gesichtspunkte nirgends auseinanderhält: ‘synthetisch’ heissen ihr solche Begriffe und Sätze, die sich nur aus Ansschauung gewinne, ‘analytisch’ solche, die sich aus reinem Denken begründen lassen. Diese Begriffsbestimmung aber enthält eine deutiliche petitio principii: sie nimmt die Entscheidung über das Problem schon in der Art der Fragestellung vorweg. Denn zweifellos kennt doch die kritische Lehre Formen der reinen intellektuellen Synthesis, die sich zwar, um empirische Erkenntnisse zu ermöglichen, auf die Anschauung con Raum und Zeit beziehen müssen, die aber ihre Wahrheit und Geltung nichtdestoweniger dem ‘reinen Verstande’ verdanken.” 188 Ch. 3 Russell on Kant

ciative student of Kant (De Laguna 1907, p. 334; emphasis in the original).

Where Cassirer is concerned with the empirical application of pure mathematics, Laguna’s point is about pure mathematics itself; since the sphere of the analytic in Kant’s original sense is the sphere of what is epistemologically unproblematic, one is no longer entitled to infer, after the realm of logic has been extended, that logic (or pure thought) coincides with the analytic in Kant’s sense. On the contrary, the new logic must be considered synthetic, if the claims which Rus- sell and Couturat and others have made on behalf of its foundational capacities are correct; and then the question, “How is pure logic possible?”, becomes acute in a way it was not for Kant. We have already seen on more than one occasion that Russell was – most of the time, anyway – well aware of these finer details of the conceptual situation. He admitted, even emphasized, that the new logic is a synthetic discipline (this point, often neglected, is duly noted by Cassirer 1907, p.37). But precisely because of its syntheticity, the new logic in general and Russell’s logicism in particular are capable of functioning significantly in a general attack on certain doctrines that are central to different forms of idealism. In the rest of this chapter, I will elaborate on the general arguments that Russell advanced against Kant (and in some cases, against idealism in general); all these arguments turn on the claim that the (transcendental)-idealist reconstruction of the synthetic apriori in fact undermines the category and its claim to constitute genuine knowledge. Once we have acquainted ourselves in more detail with Russell's conception of logic – this will be taken up in the next chapter – we are in a position to return to Russell’s criticisms of Kant and the implications that these have for his conception of logic. Ch. 3 Russell on Kant 189

3.5. The Role of Logicism

3.5.1 Hylton on the Role of Logicism in Russell’s anti- Kantianism

I shall begin this section by summarising Peter Hylton’s reconstruction of Russell’s anti-Kantian and anti-idealist arguments (Hylton 1990a, pp. 180-196; 1990b). I shall then present some reasons why this reconstruction is not fully adequate.79 These critical remarks serve as an introduction to what I take to be a more satisfactory reconstruction. As Hylton sees it, in Russell’s hands logicism constitutes the basis for a general argument against Kant and his idealist successors (Hegel, Lotze and the so-called British neo-Hegelians being prominent examples of the latter group). According to Hylton, Russell directs the argument against “a claim which he sees as crucial to any form of idealism, Kantian or post-Kantian” (1990b, p. 203). The claim is that “our ordinary knowledge (of science, history, mathematics, etc.) is, at best, true in a conditioned and non-absolute sense of truth” (id., p. 199). As Hylton points out, this straightforward general claim – based on a general contrast between appearance and reality – takes different forms for different idealists.80 For Kant, the claim constitutes the central contention of his transcendental idealism. Our knowledge is confined to the world as it appears to us; even though there are things-in- themselves, the spatiotemporal world cannot be consistently thought of as such. Kant’s idealist successors dispensed with the dichotomy

79 Here I rely extensively on Proops (2006). 80 There is the further question as to whether Kant is correctly described as an idealist at all. Hylton (1990b, p. 201), too, raises this question. Russell was in the habit of not drawing any such very precise distinctions, when it came to the views of his philosophical opponents; but I do not think that much depends on this, as long as we distinguish Kant’s position from his successors’, as Hylton does. 190 Ch. 3 Russell on Kant between the noumenal and phenomenal world. Accepting Kant’s dialectic, however, they transformed the negative claim about empirical knowledge and its object, the spatiotemporal world, into the view that our ordinary so-called knowledge cannot be regarded as adequate in the absolute sense; that knowledge in the absolute sense is attainable only by substituting idealist metaphysics with its special method and special categories of thought for conventional ways of thinking (this epistemology is accompanied by corresponding metaphysics; being contradictory, the spatiotemporal realm must be regarded as appearance as distinct from the reality, which is the ultimate subject-matter of much of idealist metaphysics).81 Russell’s logicism, Hylton argues, provides two distinct arguments, one of them explicit, the other largely implicit, against these idealist views. Firstly, and this is the explicit argument, Russell argues that modern mathematics provides means with which to refute any of the standard idealist arguments against the consistency of our ordinary ways of thinking. Central to idealism in its various forms is the contention that the categories of space and time (and anything connected with them, like change) must be written off as metaphysically illegitimate, because contradictory; these categories cannot therefore provide us with the truth about Reality.82 The most important of these

81 It goes without saying that these rather sweeping generalizations about “post-Kantian idealists” cannot lay claim to historical accuracy. For example, F. H. Bradley is conventionally ranked among “British idealists” and even “British neo-Hegelians” – he is, indeed, generally regarded as the greatest of his kin. But he did not believe that there is a special method and special categories of thought with the help of which knowledge of the Absolute (that is, Reality as opposed to Appearances) could be attained. He did not believe this, because he believed that all thought falls short of Reality. The sweeping generalization is useful, however, since it pinpoints a line of thought that is, arguably, relevant for understanding Russell’s reaction to his philosophical predecessors. 82 According to post-Kantian idealists, ordinary categories, though contradictory, are not completely useless for that; for they may be capable of unfolding partial truth about reality, if they are regarded as necessary steps in the dialectic towards a metaphysical picture of the world. Ch. 3 Russell on Kant 191 arguments are the antinomies with which Kant claimed to be able to prove that if space and time are taken as things-in-themselves or attributes thereof, then contradictions inevitably follow.83 Russell’s counter-argument is that difficulties like Kant’s antinomies are resolved by “modern mathematics”. Russell, Hylton argues, “claims that modern mathematics makes available consistent theories which may represent the truth about space and time; whether they in fact do so is a matter on which he is willing to remain agnostic. The crucial point is that mathematics makes consistent theories of space and time available” (1990b, p. 202). Here, of course, Russell draws on the work done by Cantor, Dedekind and Weierstrass. But, Hylton adds, this anti-idealist argument depends crucially on logicism, too: “[i]t is important, however, to see that it also depends upon the central claim of Russell’s logicism, that mathematics is wholly independent of the Kantian forms of intuition. It is only if mathematics is in this way independent of space and time that it can be used, in non- circular fashion, as an argument for the consistency of the latter notions” (id., 1990a, p. 180).

83 More precisely, Kant argued that he could prove, for certain pairs of attributes, two apparently contradictory propositions concerning the world as a whole. “[T]his great conflict of reason with itself” (Allison 1983, p. 36) is resolved by noting that the propositions in question are contradictory only on the assumption that they are about the world as a thing-in-itself; if this assumption is given up – if, that is, the world as a whole is considered transcendentally ideal, or not independent of the knowing subject or conditions of (human) knowledge – then the antinomies disappear, as the propositions are seen not to be really contradictory. For instance, the thesis of Kant’s first antinomy says that the world has a beginning in time and a limit in space, and the antithesis denies this, holding that the world is both spatially and temporally infinite. Kant now argues that there are compelling reductio proofs for both the thesis and the antithesis. This yields the conflict of reason with itself. It can be resolved, however, by noting that finitude and infinitude are predicates that attach to the world only on the assumption that the world exists in itself. If the assumption of independent existence is given up, then it becomes understandable, according to Kant, how the world can be neither finite nor infinite. 192 Ch. 3 Russell on Kant

Russell has another anti-idealist argument to offer, though it remains largely implicit, according to Hylton. The second argument presents (pure) mathematics “as a particularly clear counterexample” (1990b, p. 203) to the idealist conception of ordinary knowledge. (Hylton does not insert the qualification “pure”, but it is clearly needed, for Russell is concerned, throughout the Principles, with pure mathematics only.) For idealists, ordinary knowledge and, more generally, ordinary categories of thought are conditioned or non-absolute in one way or another. Hylton explains the Kantian version of this view as follows: “[a] direct consequence of the Kantian version of the claim is that our knowledge is confined to what can be given in intuition, i.e. to actual and possible objects of sense experience. Since these objects are partially constituted by our minds, a second consequence of the Kantian view is that our knowledge is conditioned by the nature of our cognitive faculties” (ibid.) The post-Kantian idealist in turn argues that our ordinary ways of knowledge are at best relatively true. To undermine these claims, Russell sets out to show that pure mathematics is true in a simple and straightforward “Moorean” sense. As Hylton puts it, the logicist Russell sets out to show “that mathematics is true – not true just as one stage in dialectic, or more or less true, but true absolutely and unconditionally; not just true if put in a wider context, or if seen as part of a larger whole, but true just as it stands; not, to revert to the Kantian idiom, true from the empirical standpoint but false from the transcendental standpoint, but simply TRUE, with no distinctions of standpoint accepted” (ibid.) Hylton argues that here the central implication of logicism, namely that since logic is independent of space and time and mathematics is reducible to logic, mathematics, too, is similarly independent, is important for two reasons. Firstly, if mathematics were based on space and time, then all the idealist doubts about these categories would accrue to mathematics as well. Secondly, if mathematics were dependent of space and time, then it would not be unconditionally true; its truth would be limited to the spatiotemporal sphere (ibid.) Ch. 3 Russell on Kant 193

3.5.2 Criticism of Hylton’s Reconstruction

This, in outline, is Hylton’s reconstruction of Russell’s general anti- idealist arguments. I shall next consider it in more detail. Let us begin with the explicit argument that Hylton attributes to Russell. It is undeniable that Russell did subscribe to some such argument. The following passage from the Principles constitutes sufficient evidence:84

The questions of chief importance to us, as regards the Kantian theory, are two, namely (1) are the reasonings in mathematics in any way different from those of Formal Logic? (2) are there any contradictions in the notions of time and space? If these two pillars of the Kantian edifice can be pulled down, we shall have successfully played the part of Samson towards his disciples. (Russell 1903a, §433)

Unfortunately, it is not clear exactly what argument Hylton intends to attribute to Russell. On the face of it, it is the existence of consistent theories of space and time (“theories that may represent the truth about space and time” (Hylton 1990a, p. 181)) that Russell uses as an argument against idealists. Hylton suggests, however, that any straightforward argument from “modern mathematics” (Weierstrass, Dedekind, Cantor) is potentially vulnerable to a charge of circularity. This suggestion may be turned into the following explicit argument:

1) modern mathematics (in particular, the arithmetization of the continuum) has made available consistent theories of space and time; 2) these theories make use of notions that themselves presuppose spatial and/or temporal notions (potential idealist retort to 1)); 3) therefore, these theories themselves do not suffice to show that spatial and temporal notions are free from the difficulties that idealists have traditionally found in them.

84 Hylton, too, quotes this passage; Hylton (1990a, pp. 181-2), (1990b, p. 202). 194 Ch. 3 Russell on Kant

Logicism is then brought to bear on the second premise; since logic is independent of space and time, logicism shows that mathematics, too, is thus independent; hence the charge of circularity can be circumvented. This is the first and, in my opinion, the most plausible construal of the argument that Hylton attributes to Russell. On this view, what worries Russell is the potential dependence of the relevant parts of mathematics on spatiotemporal notions, and this dependence is shown to be non-existent by the logicist reduction. This is not the only possible interpretation of Hylton’s Russell, however. For it may be that Hylton sees Russell as having been concerned not only about the worrisome dependence but also about the consistency of the relevant pieces of mathematics (the relevant text runs as follows: “[i]t is only if mathematics is in this way independent of space and time that it can be used, in non-circular fashion, as an argument for the consistency of the latter notions” (Hylton 1990a, p. 181, 1990b, p. 202; emphasis added). That this is Hylton’s view is suggested in Proops (2006):

Hylton’s presentation is compressed and it is hard to be certain how the details of the argument are supposed to unfold, but we may perhaps construct a cogent argument along the line he suggests, at least for the case of time. Plausibly, in order to establish the consistency of the concept of time, Russell would need to formulate a theory of time and show that this theory had a model. So one might have expected him to formulate an axiomatized theory of time (as, say, Dedekind complete, dense linear ordering without end points) and then to find a model for this theory. If he had thought of the real numbers as providing such a model, he might then have worried that the theory of reals could itself be essentially dependent on time, and so still potentially inconsistent. Had he reasoned this way, Russell might have appealed to the logicist reduction of the theory of the reals in order, in effect, to provide a relative consistency proof for this theory. (id., p. 281) Proops is hesitant to endorse this argument as an exegesis of Russell, and I concur with this judgment. We must nevertheless acknowledge that Russell himself does use such terms as “non-contradictory” and “logically permissible” in this connection, something that neither Hyl- Ch. 3 Russell on Kant 195 ton nor Proops points out. I have found two such passages. The first is in an essay entitled “Is Position in Time and Space Absolute or Relative?” which Russell composed in 1901 and parts of which he incorporated into the Principles. He explains the aim of the paper as follows:

I shall not be concerned directly with the existence of space and time, but only with their logical analysis: that is, I shall not raise the question whether what appears to exist in space and time is mere appearance, but only the question what space and time must be if what appears to occupy them does exist. It should be observed, however, that, if I succeed in constructing a logically permissible account of space and time, the chief reason for regarding their content as mere appearance is thereby destroyed. (1901c, p. 261; emphasis added)

The second passage is, admittedly, from a later period:

The paradoxes of Zeno the Eleatic and the difficulties in the analysis of space, of time, and of motion, are all completely explained by means of the modern theory of continuity. This is because a non-contradictory theory has been found, according to which the continuum is composed of an infinity of distinct elements; and this formerly appeared impossible. The elements cannot all be reached by continual dichotomy, but it does not follow that these elements do not exist. From this follows a complete revolution in the philosophy of space and time. The realist theories which were believed to be contradictory are so no longer, and the idealist theories have lost any excuse there might have been for their existence. (1911a, p. 34; emphasis added)

Despite such passages, the point behind Russell’s talk about logically permissible and non-contradictory theories is not really about consistency-proofs. As Proops points out,85 there is never an indication in Russell that a positive argument for consistency is either necessary or attainable. Russell’s general argument for the consistency of such notions as continuity and infinity is simply to rebut in a piecemeal fash-

85 Proops (2006, pp. 281-2). 196 Ch. 3 Russell on Kant ion certain traditional arguments, like those presented by Zeno and Kant; with these arguments dismissed, he believes that there are no other cogent reasons to doubt the consistency of the spatiotemporal continuum. In particular, then, there is no need to invoke logicism and consistency-proofs to establish that the relevant notions are free of contradiction. This argumentative strategy is seen at work in chapter 43 of the Principles, where Russell discusses the question whether “any contradiction can be found in the notion of the infinite”. Russell praises Kant for his attempt to have exhibited precise contradictions in the notion of the infinite, a task that philosophers, according to Russell, have as a rule shunned. Russell argues, in general, that all attempts by previous (that is, roughly, pre-Cantorian) thinkers to elicit contradictions from the notion of the infinite have been based on the alleged difficulty with the notion of one-to-one correspondence between whole and part (1903a, §337). The pre-Cantorians had in fact assumed, in one form or another, the axiom that “the whole cannot be similar to the part” (id., §341). With the help of this axiom, precise contradictions can be elicited, and the axiom does have common sense on its side (ibid., see also Russell (1901a, p 373)). But there is no further evidence for the axiom except its supposed self-evidence. Common sense and self-evidence are outweighed by the consideration that the consequences of the axiom are destructive for mathematics; when, however, the axiom is rejected, everything goes well. Thus, Russell suggests, the primary obstacle in the way of recognising the actual infinite is, broadly speaking, a psychological one. In our daily lives – “in engineering, astronomy, or accounts, even those of Rockefeller and the Chancellor of the Exchequer” (1903a, §341) – we are accustomed to dealing with finite numbers, and here it is provable that there can be no such similarity or one-to-one correspondence. Hence the felt impossibility is easily explained, but none the less lacks an adequate mathematical foundation. It seems, then, that an adequate and sufficient defence of the infinite consists, therefore, of an argument which dismisses “the usual objections” against the infinite as “groundless” (id., §344). Ch. 3 Russell on Kant 197

I conclude that the second construal of Russell’s explicit argument against idealism is not plausible. We must next consider the first alternative. According to it, logicism is needed to circumvent the charge of circularity by demonstrating that mathematics is independent of space and time. This second suggestion, however, is no more attractive than the first. For it is clear that Russell does not think that logicism is needed to show that mathematics is independent in the required way; the idea that mathematical reasoning (in the broad sense) somehow involves space and time is shown to be erroneous by the arithmetization of mathematics. He says this much in section 249 of the Principles, from which I have already quoted (see section 3.3.3). The point is that modern mathematics has shown that continuity has no essential reference to space and time and that the concepts of the Calculus do not presuppose motion or change. This recognition that the “fundamental problem of mathematical philosophy”, i.e., the problem of infinity and continuity, is already present in “pure arithmetic” is due to the labours of Weierstrass and Cantor (ibid.) And it is they who have effected a complete transformation in mathematicians’ understanding of this problem; logicism is therefore not needed here. There is no other plausible construal of the explicit argument that Hylton attributes to Russell. I conclude, therefore, that even though Hylton is right when he credits Russell with the claim that “modern mathematics” has conclusively refuted certain idealist arguments regarding space and time, he has not correctly delineated Russell’s reasons for saying so. More precisely, I do not think – and here I agree with Proops – that logicism plays any positive role in this particular argument by Russell; he disputes idealists’ arguments against space and time simply on the grounds that they are mathematically unsound, and this in turn is shown by modern treatments of infinity and continuity. We must next consider the implicit argument that Hylton attributes to Russell. Here, I submit, we find a more promising candidate for an anti-Kantian, and even anti-idealist, argument that is based on logicist premises. Recall that, according to Hylton, Russell presents (pure) mathematics as a counterexample to the crucial idealist claim 198 Ch. 3 Russell on Kant that what ordinarily counts as knowledge is not true in the absolute and unconditional sense. Again, some care must be exercised in identifying the argument of which this claim could be the conclusion. Since Hylton at most suggests that there is such an argument but does not formulate it explicitly, I start from the proposal made by Proops (2006) on Hylton’s behalf: “He [sc. Hylton] makes a number of remarks that suggest the following reasoning: It is accepted on all sides that logic, at least, is unconditionally and absolutely true. Logi- cism then extends the unconditional and absolute truth of logic to one branch of ‘ordinary’ human knowledge – namely, to pure mathematics. This generates a counterexample to the ‘crucial idealist claim’ about truth”.86 Proops argues that the attribution of this argument to Russell faces two powerful objections. Firstly, “there is simply no indication in the text that this is Russell’s intended strategy”. Secondly, he claims that “it is doubtful that Russell could have intended to argue this way.” Of these two points, I shall set aside the first one. After all, Hylton admits that the argument he is attributing to Russell is only implicitly there; if this is so, it does not seem to matter much if there is no direct evidence for the attribution – passages in which Russell argues in the manner described. Proops’ second point is therefore more important. He detects the following difficulty in the suggestion that the reconstructed argument was really Russell’s line of reasoning: “because the argument is intended to have suasive force against idealists, they too must be prepared to grant its assumption that logic is unconditionally true. But, although Kant might have accepted this assumption, it [is] unclear that post-Kantian idealists in general would have done so”.87 What Proops has to say about post-Kantian idealists is completely correct. No doubt, they were unwilling to admit that anything deserv-

86 Proops (2006, p.279); italics in the original. 87 Proops (2006, p. 279); italics in the original. Ch. 3 Russell on Kant 199 ing the title “formal logic” was capable of unconditional or metaphysical truth.88

88 Proops (2006, pp. 279-80) uses Bradley to illustrate this claim. Bradley has two criteria for something’s being “more true/less false”: being (a) more concrete and (b) more extensive in application (see Bradley 1893, Ch. XXIV). These criteria point in opposite directions. The truths of logic are plausibly the most abstract truths there are: hence they cannot possess any great degree of truth by the first criterion. On the other hand, it is natural to think that the more abstract a truth is, the more extensive it is in its application; by this criterion, logic should be more rather than less true. Proops concludes from this that the picture is blurred. This, however, is not the conclusion that we should draw from Bradley’s discussion. The two criteria are needed, Bradley argues, because we must take into account, in our attempts to estimate the degree of reality of a thing, both events (occupants of space and time) and laws. Comparing these two, we get opposite results. Since laws are abstract, general truths, they are “remote from fact, more empty and incapable of self-existence” and, therefore, less true by the first criterion. On the other hand, “the more concrete connexions of life and mind”, being valid for more limited stretches of reality, are less true by the second criterion. So, they do point in opposite directions. But, contrary to what Proops suggests, the picture is not really that blurred, since both (a) and (b) are essential to Bradley’s conception of reality; indeed, qua criteria of reality, they flow directly from his conception of the Absolute. The briefest possible description of the Bradleian Absolute is this: “the Absolute is [...] an individual and a system” (id., p. 127); more precisely, it is “a single and all- inclusive experience” (id., p. 129). The degree of reality of a thing is determined by the degree to which it can be likened to the Absolute. Hence, the degree of reality is determined by the extent to which it is a true individual, capable of self-existence, and incorporating everything into a harmonious system, a genuine unity. When we look at things, so to speak, from the standpoint of the Absolute, we can see that the two criteria are not really two at all. Being one and being a system are one and the same thing. If something is really one, it does not exclude anything, but has everything contained in itself in a “harmonious” and consistent system, a genuine unity. As regards the truth of logic, they are judged deficient by criterion (a), and cannot therefore deliver truth in the absolute, metaphysical sense. That they should then turn out to be more, rather than less, true by criterion (b) is 200 Ch. 3 Russell on Kant

Such a view is slightly more plausible in Kant’s case. For Kant held the view that formal logic is not subject to the restrictions that he detected in object-related human thought; hence the idea of a thing-in-itself is consistently thinkable in the very thin sense licensed by formal (general) logic. However, since Russell’s formal logic has, at least extensionally, very little to do with Kant’s, it remains unclear whether an argument based on the former could have suasive force against Kant’s position. Fortunately, we need not resolve this question here. For it seems fairly clear that Hylton does not in fact intend the argument to be suasive in the way suggested by Proops. This much is indicated by what Hylton has to say about the general argumentative shape of Russell’s dialectics vis-a-vis the idealists (see Hylton 1990a, pp. 115- 116). Hylton acknowledges that Russell’s use of the new logic in his anti-idealist argument is circular (id. p. 116). On the one hand, Russell thinks that the new logic shows that mathematics is not inconsistent, which undermines idealism; on the other hand, the new logic – at least the use to which Russell wants to put it – makes certain metaphysical demands on such notions as truth and knowledge, and these demands are at odds with the idealist construal of these notions. Rus- sell’s metaphysics, then, is flatly incompatible with idealism. Thus, if one is not already disposed against idealism, “the new logic will not appear to have the status which it must have if it is really to show that mathematics is fully coherent and consistent” (ibid.) This admission of circularity shows that what Hylton has in mind is not the suasive argument that Proops ascribes to him. As I read him, Hylton intends to ascribe the following argument to Russell:

perhaps only to be expected, since “all thought is to some degree true”, according to Bradley (id., p. 321). Ch. 3 Russell on Kant 201

(*) The new logic is independent of those restrictions that Kant and other idealist impose on (ordinary) human cognition. The new logic is nevertheless capable of grounding genuine claims to knowledge (a claim demonstrated by logicism). Therefore, there is at least one region of human knowledge, viz. pure mathematics, which is not subject to the limitations that idealists claim to find in all non-metaphysical thought.

(*) has no suasive force against idealists. The argument simply assumes what idealists deny, namely that there are regions of ordinary knowledge which are capable of genuine (absolute, unconditional, metaphysical) truth. The intended audience to which (*) is addressed consists of people who are untainted by idealist metaphysics and who, therefore, believe in truth in the ordinary sense, which is the absolute, un-watered-down sense of science and common sense. On the other hand, Russell is also in a position to argue that the fact that the metaphysics which he has learned from Moore and which underlies logicism allows mathematics to be true is a powerful argument for accepting that metaphysics (see Russell 1903a, p. xviii). It may well be that (*) is unnecessarily complicated as an interpretation of Russell. Hylton is correct in arguing that Russell thought that there is a crucial difference between his position and the idealist one, as regards the absolute and unconditional character of knowledge and truth. It is one of the chief defects of any sort of idealism, according to Russell, that it does not allow mathematics to be absolutely and unconditionally true; hence it does not allow mathematics to constitute real knowledge. But it is less clear that he held the view that logic is needed to guarantee the absolute character of mathematical knowledge and truth. Rather – and here I am inclined to agree with Proops – Russell takes these characteristics as an unquestionable given, and argues that any philosophy which denies them must be rejected for that reason, idealism in all of its forms being the prime example of such denial. In the Preface to Principles, the point is formulated as follows: “Formally, my premisses are simply assumed; but the fact they allow mathematics to be true, which most current phi- 202 Ch. 3 Russell on Kant losophies do not, is surely a powerful argument in their favour” (1903a, p. xviii). The premises to which Russell is referring here are the metaphysical views which he says he has learned from Moore. The claim is then that the fact that Moorean metaphysics allows mathematics to be true is a powerful argument for accepting it, and rejecting anything that is incompatible with, like its idealist competi- tors. Although (*) comes at least close to being genuinely Russellian, it is probably advisable to give logic an explanatory role only. That is to say, Russell is probably best seen as arguing that idealism, be it Kant- ian or post-Kantian, does not allow the attribution to mathematics of certain properties (including the properties that mathematical knowledge is genuine knowledge and mathematical truth is genuine, absolute, unconditional, or “metaphysical” truth) which are correctly thought to belong to it. Logicism together with its underlying conception of logic are then invoked as the true explanation of why these properties do attach to mathematics; this explanation is of course presented as an alternative to idealist explanations, of which Kant’s theory of the synthetic apriori is the best-known and clearest. This is the line of thought that I shall focus on in the remaining sections of this chapter, and I shall develop its consequences by relating it, precisely, to Kant’s philosophy. The reason for making this limitation is that the contrast between Russell and idealism is clearest in the case of Kant. It is to be noted also that thoughts about formal logic play a positive role in Kant’s philosophy (as was pointed out above, here there is a conspicuous difference between Kant and his idealist successors); what he has to say about this topic constitutes a useful background for a detailed discussion of Russell’s early conception of logic. Ch. 3 Russell on Kant 203

3.6. Russell’s Case against Kant

3.6.1 The Standard Picture of Transcendental Idealism

In the previous section the conclusion was reached that the main difference between Russell and Kant is to be found in the different explanatory strategies they apply to the problematic category of synthetic apriori truths. To quote from the Introduction to this chapter, Russell thinks that Kant’s theory of mathematics is defective in that it misrepresents or compromises the senses of those attributes which it – correctly – ascribes to mathematics. The starting-point is the view that mathematics is apriori. Now, according to long philosophical tradition about apriority, this implies three further properties: truth, necessity and universality. According to Russell, Kant’s theory of the synthetic apriori fails to do justice to these properties: Kant’s theory misrepresents the sense in which the propositions of mathematics are true and universal; and it does not make them genuinely necessary, either. It must be added, though, that the last of these attributes turns out to be problematic in view of Russell’s somewhat ambivalent attitude towards modal notions. The general point remains, however, that in Russell’s view, his own philosophy of mathematics succeeds better than does transcendental idealism in explaining how the propositions of mathematics can have a number of important properties. And a failure in this respect is something that he sees as a powerful argument against transcendental idealism. Russell’s criticisms of Kant and his explanatory strategy stem from a specific interpretation of transcendental idealism. It is a fairly typical example of what Henry Allison had dubbed the standard picture. Represented by such well-known Kant-interpreters as H. A. Prichard and Peter Strawson, the standard picture can be summarized by the following seven statements. These are derived from Allison’s discussion (1983, pp. 3-6); I insert in footnotes a few quotations from Rus- sell, in order to make it plausible that the standard picture was also his: 204 Ch. 3 Russell on Kant

(1) “Kant’s transcendental idealism is a metaphysical theory that affirms the unknowability of the ‘real’ (things in themselves) and relegates knowledge to the purely subjective realm of representations (appearances).” (Hylton 1983, pp. 3-4) (2) This postulation of things in themselves, though clearly problematic, is “deemed necessary to explain how the mind acquires its representations or at least the material for them” (id., p. 4). (3) “The basic assumption is [...] that the mind can only acquire these materials as a result of being ‘affected’ by things in themselves. Thus, such entities must be assumed to exist, even though the theory denies that we have any right to say anything about them (ibid.)”89 (4) Transcendental idealism is a direct consequence of the contrast between a realm of physical objects composed of primary qualities and a mental realm consisting of the sensible appearances of these objects; the mental realm, or the realm of representations, is produced by means of an affection of the mind by physical objects.90 (5) Kant’s version of (4) is his doctrine which assigns the entire spatiotemporal framework to the subjective constitution of the human mind.91 The most basic objection to Kant which

89 “That Kant was able to assume even an unknowable thing-in-itself was only due to his extension of cause (and ground) beyond experience, by regarding something not ourselves as the source of our perceptions.” (Russell 1900, p. 74) 90 This formulation is adopted from Strawson (1966, pp. 39-42). Russell comes very close to it in (1912a, p. 48). 91 “What is distinctive of Kant is the way he apportions the shares of ourselves and the physical object respectively. He considers that the crude material given in sensation […] is due to the object, and that what we supply is the arrangement in space and time, and all the relations between sense- data which result from comparison or from considering one as the cause of the other or in any other way.” (Russell 1912a, p. 48) Ch. 3 Russell on Kant 205

emerges from the standard picture is the claim that by limiting knowledge to appearance – to the subjective realm of representations – he effectively undermines the possibility of our possessing any genuine knowledge at all. (6) The sharpest formulation of (5) is to construe Kant’s distinction between appearances and things in themselves as a case of perceptual illusion.92 (7) Given the construal mentioned in (6), there is little difficulty in reducing Kant’s theory to absurdity; the doctrine that we only know appearances means that “we can know things only as they ‘are for us’ or ‘seem to us’ (in virtue of the distortion imposed by our perceptual forms), not as they ‘really are’. Since to know something just means to know it as it really is, it follows that for Kant we cannot really know anything at all” (Allison 1983, p. 6).93

As the quotations given in the footnotes indicate, Russell’s commitment to the standard picture is clearest after his adoption of the sense-data analysis of perception. Elaborating on points (4) and (5) made above – and thus on Strawson’s (1966) analysis of the conceptual situation – we may say that Kant shares with a more orthodox sense-data theorist like Russell a common starting-point, namely the contrast between the realm of objects of which we are aware as a consequence of being affected by them – things that appear to us by affecting us – and the way they appear to us.

92 This is Prichard’s formulation. He illustrates Kant’s distinction with the help of the analogy of a straight stick that appears bent to an observer when it is immersed in water (Prichard 1909, pp. 72-75). Russell uses the classical coloured spectacles -example to the same effect; see Russell (1911a, p. 39). 93 This was also Russell’s view: “Space and time and the categories inter- pose a mirage of illusion which cannot be penetrated at any point. As an answer to Hume’s scepticism, this seems a somewhat unsuccessful effort” (1927, p. 199). 206 Ch. 3 Russell on Kant

According to Strawson, this contrast is intelligible, though perhaps not very attractive, as long as it is construed as the distinction between physical things and their appearances. He maintains, however, that Kant’s version of the theory transgresses the bounds of intelligibility; this is due to the peculiarly Kantian assignment of the entire spatiotemporal framework to the mind, or the “receptive constitution of the subject of experience” (1966, p. 41). For Kant’s strategy takes the notion of an “object affecting the subject” outside the spatiotemporal range, which is the proper range of its intelligible employment. Kant’s way of effecting the division between the respective contributions of “objects” and “ourselves” makes the doctrine impossible to understand; since we are no longer speaking of what is in space and time, we do not know what “affecting” and “ourselves” are supposed to mean. The second difficulty with Kant’s doctrine – and this is also one of Russell’s criticisms – is that it deprives us of all empirical knowledge. Unlike the orthodox sense-data theorist, Kant is in no position to hold that we acquire knowledge of a certain range of things, namely physical objects, through affection; rather, the doctrine leads to an emphatic denial of knowledge. The spatiotemporal field is no longer something that we can have knowledge of (albeit only indi- rectly); on the contrary, things ordered in space and time become mere appearances standing forever between ourselves and reality, or things in themselves, and forging an unbridgeable gap between the two. The briefest possible description of the standard picture is to say that it represents transcendental idealism as a combination of phenomenalism not unlike that of Berkeley with the somewhat dubious postulation of things in themselves. It is an immediate consequence of this that Kant is regarded as transforming ordinary things into mere representations, while things in themselves or things as they really are become the unknowable something that is supposed to give rise to appearances in some way. Ch. 3 Russell on Kant 207

3.6.2 The Implications of the Standard Picture

3.6.2.1 Kant’s “Subjectivism”

Given the standard picture, it is not difficult to see why Russell should have thought that the most fundamental thing that is wrong with transcendental idealism is its irremediable subjectivity. This allegation is particularly pregnant in the case of apriori knowledge. For even if it could be argued that Kant can retain an element of objectivity for aposteriori knowledge – this would be “the crude material given in sensation” that Russell mentions in (1912a, p. 48), although it is not clear that Kant is entitled to even that – his explanatory strategy for apriori knowledge seems to have the consequence that such knowledge is concerned exclusively with characteristics that are, in some sense, due to our own nature. There are many ways of interpreting such expressions as “due to our nature”. In the Kantian context such phrases tend to become ambiguous in a way that can only be described as dangerous. Russell, however, was inclined to mean by “subjective” what most people mean by it, the result being that Kant’s account of knowledge seemed irredeemably subjective (or non-objective) to him. His dislike for “subjectivism” in knowledge was very strong, and the charge that Kant’s theory in general and his explanatory strategy for the apriori in particular is vitiated by it was not restricted to the sense-data period. Decades later he reminisced that in 1898 various things caused him to reject Kant and Hegel, and he mentions, in particular, that he dislike the subjectivity of the Transcendental Aesthetic (1944, pp. 11-12). In his Leibniz-book the crucial point is formulated by saying that Kant’s theory of experience renders all apriori knowledge “self-knowledge” (1900, p. 74). The following two quotations are representative of the early Russell’s reading of Kant:

Thus Leibniz is forced, in order to maintain the subject-predicate doctrine, to the Kantian theory that relations, though veritable, are the work of the mind. [...] The view, implied in this theory and constituting a large 208 Ch. 3 Russell on Kant

part of Kant’s Copernican revolution, that propositions may acquire truth by being believed, will be criticized in connection with the deduction of God’s existence from the eternal truths. (id., p. 14)

Lotze presumably holds that the mind is in some sense creative – that what it intuits acquires, in some sense, an existence which it would not have if it were not intuited. Some such theory is essential to every form of Kantianism – to the belief, that is, that propositions which are believed solely because the mind is so made that we cannot but believe them may yet be true in virtue of our belief. (1903a, §427)

Although the charge of subjectivity is not mentioned by name in these passages, it is closely connected with the claim that Kant’s theory of the apriori is concerned, in some sense, with the “contributions of the mind”, there being no other than a “subjectivist” reading of this notion for Russell.94 The passages quoted above are about the most elaborate descriptions that the early realist Russell gives of transcendental idealism. They can scarcely be regarded as self- explanatory; for example, why would anyone want to say that Kant believed that “propositions may acquire truth by being believed”? Does this really constitute a large part of the Copernican revolution, as Russell suggests?95

3.6.2.2 Moore against Kant

It is important to see both what, exactly, the view is that Russell is attributing to Kant in passages like these and why he interprets Kant in this admittedly non-standard manner. Since Russell himself is rather reticent about his reading of Kant, we must draw on what other advocates of the standard picture have said. Particularly helpful

94 This charge occurs repeatedly in Russell’s book on Leibniz: see Russell (1900, pp. 99, 119, 157, 163, 181). 95 And Russell himself admits that this is a non-standard formulation of Kant’s views (1900, p. 14, fn. 2). Ch. 3 Russell on Kant 209 in this connection is Moore’s early essay Kant’s Idealism.96 As we shall see, most of the points that Moore makes have close analogues in Russell’s often very terse criticisms of Kant. Transcendental idealism, Moore explains, is Kant’s answer to the question, How can we know universal synthetic propositions to be true? These are propositions of the form “All instances of A have the predicate B” or “Anything which has the predicate A has the predicate B” (1903-4, p. 130). The answer itself to the “how can we know”-question (or one particular case of it) is this: “[i]t is only if the mind is so constituted that, whenever anything is presented to it, it invests that thing with geometrical properties, that we can be entitled to assert that everything we shall ever experience will have those properties” (id., p. 129). Thus, “the only predicates which do attach to all things – formal predicates – are given to them by the mind” (ibid.) This is part of what Kant means to assert by his transcendental idealism, according to Moore. Next, however, a distinction is brought to bear on the discussion that gives it a twist that is typical of the realist and anti-idealist Moore. For he explains that Kant’s question – How are synthetic propositions a priori possible? – is ambiguous, because it is a general rule that two questions are asked, whenever we ask: How do we know a thing? Since the concept of knowledge is a complex one, we mean two distinct things when we say that we know something: we mean that we believe it, that is, that we have a certain mental attitude towards it, and we also mean that what we believe is true. It follows that in asking “How do you know?” we may be inquiring into the causes of the person’s beliefs (How do you come to believe?), or else we may be interested in the reasons he or she has for the belief (How do you know that what you believe is true?). Moore does not claim that Kant is simply confused about this distinction or that he fails to draw it. On the contrary, he openly acknowledges that what Kant had in mind was the question of validity: “he [sc. Kant] wished to explain the validity of universal propositions; not only how we could come to

96 Moore (1903-04). 210 Ch. 3 Russell on Kant believe them, but how they could be valid. Only so could he be con- tradicting Hume’s sceptical conclusion. Hume asserted: We have no title to believe that every event has a cause; and Kant answers: We have a title; I can prove it true that every event has a cause” (1903-4, p. 132)). He does claim, however, that Kant’s explanatory strategy is inadequate; that a universal proposition’s being a “condition of the possibility of experience” can do nothing to establish the validity of such propositions. Moore has three distinct arguments to establish this conclusion. Firstly, there is the charge that Kant’s argument with which he tries to establish the validity of certain universal propositions is dependent upon a premise that is itself a universal proposition: Kant’s answer to the question, “How do we know certain universal propositions to be true?” is to say that “[m]ind always acts in a certain way upon, arranges in a certain manner, everything which is presented to it” (id., p. 133). At the very least, then, Kant’s strategy cannot be accepted as valid for all synthetic universal propositions.97 The second argument is that what Kant in fact succeeds in proving is not the original proposition which he professes to prove but something much weaker: what the argument can establish is not that every thing has such and such properties but at most that every thing, when it is presented to the mind, has these properties; as Moore himself puts it, “Kant’s premiss does not prove that 2 + 2 = 4 in every case: on the contrary, it allows that more often than not 2 + 2 may make 5 or any other number” (id., p. 134; emphasis in the original). And even this conclusion grants too much to Kant, according to Moore. To show this, he introduces his third argument, and this is the most important in this connection; or at any rate, its conclusion is. Moore suggests that Kant is in fact guilty of a confusion: he fails to distinguish the idea that the mind gives objects certain properties from the entirely different idea that “the nature of our mind causes us to think that one thing causes another, and to think that 2 and 2 are 4”

97 This argument is also developed in Moore’s lectures which were ater published as Some Main Problems of Philosophy; see Moore (1953, pp. 150-4). Ch. 3 Russell on Kant 211

(id., p. 135). Of these two ideas the former is the more favourable one to Kant. For suppose that the mind really is capable of “imposing” certain properties on objects. And suppose also that this imposition is an activity that does take place. It follows that objects really and truly have these properties. In other words, even if, as Moore argues, Kant could only prove for a certain specific subset of absolutely all things that the mind imposes certain “formal” predicates on them, he would still be entitled to hold that the members of this subset are truly characterized by these predicates. But the very idea of mind giving properties to things is one that Moore has difficulties in comprehending; in fact he goes so far as to write that “[n]o one, I think, has ever definitely maintained the proposition that the mind actually gives properties to things: that, e.g., it makes one thing cause another, or makes 2 and 2 = 4” (ibid.) And so he suggests that Kant, too, was capable of entertaining such an idea only because he confused it with the more plausible one that our mind is so constituted as to make us think in a certain way.98 If this more plausible idea is made part of transcendental idealism, however, then the conclusion must be drawn that Kant’s explanatory

98 Moore argues there is another reason for thinking that the idea that the mind makes us think in a particular way was really part of Kant’s transcendental idealism. He refers to the notion of a thing in itself, explaining that Kant’s view was that we can never know of anything what it is in itself, that I never know, for example, that the fingers in my right hand, as they are in themselves, are really five in number. Moore’s point (which is similar to one that Prichard made (1909, pp. 77-78)) is that since “to know a thing” means to know that thing as it is in itself, Kant’s view – and any other view which holds that we do not know things as they are in themselves – has the consequence that “we have no knowledge at all” (1903-04, p. 136). And, he adds, this conclusion cannot be sidestepped by drawing a contrast between “things in themselves” and “objects of experience”: for even to know an object of experience is to know a thing as it is in itself. Thus, Moore concludes, Kant’s denial that we have knowledge of things-in-themselves gives us a reason to think that he did not really believe that the mind gives properties to objects, but merely that it makes us think that those objects have these properties. 212 Ch. 3 Russell on Kant strategy is singularly unsuccessful as an answer to the disambiguated version of the original question, i.e., to the question, “What warrant do we have for synthetic universal propositions?” For clearly the proposition that the nature of our mind makes us think that objects have certain properties is no warrant for the claim that objects in fact have such properties; such a proposition could at most be a description of the causes of our beliefs; as Moore puts it, “[f]rom the fact that we always think a thing it certainly does not follow that what we think is true” (id., p. 136).

3.6.3 The Relativized Model of the Apriori

3.6.3.1 Preliminary Remarks

I shall give the name the relativized model of the apriori (r-model for short) to Kant’s model of the apriori, as it was interpreted by Moore (and Russell). According to the r-model, the gist of transcendental idealism is the idea that all our apriori knowledge is grounded in certain standing features of the human mind – somewhat in the manner of the logical empiricists, who argued that the apriori is based, in some not too clear sense, on linguistic conventions. Thus, Kant’s is not the only relativized model of the apriori. Since, however, there is no risk of confusion here, I shall stick to the simple label. It seems clear enough that Russell’s talk about “propositions which acquire truth by being believed” is to be understood in accordance with the r-model. The propositions in question are the different kinds of synthetic apriori propositions whose possibility Kant is trying to explain; and talk of their acquiring truth by being believed or being true in virtue of our beliefs is a way – maybe somewhat fanciful – of indicating that they enjoy a special status: they are psychologically necessary propositions (or, perhaps, consequences of such propositions, in which case they qualify as psychologically necessary in a derivative sense). That is, they are propositions which we cannot but believe, given the constitution of our minds; they are nevertheless Ch. 3 Russell on Kant 213 true, and objectively so, and not merely figments produced by our mental constitution.99 It is far from obvious how these two features, psychological necessity and objective truth, are to be reconciled, or how the r-model is supposed to effect the reconciliation. This, of course, is precisely the point that Moore and Russell focus on in their criticisms of Kant. And as our previous discussion of Moore’s views on Kant shows, the conclusion is not far-fetched that no such reconciliation is in fact possible – in particular when the problem is formulated in the rather bold manner that Moore and Russell tend to assume in their discussion of Kant and transcendental idealism.

3.6.3.2 Three Direct Arguments Against the R-model

Above we saw Moore arguing that Kant was simply confused about the idea – arguably central to any interpretation of transcendental idealism – that the mind imposes certain predicates or properties on objects. He claimed, in fact, that this idea defies comprehension, thereby suggesting that not even Kant meant that this idea should be taken literally. This argument (not really that untypical of Moore) is a particularly straightforward attempt to show that Kant’s position is somehow incoherent or even unintelligible. From Moore’s and Rus- sell’s writings we can extract two other direct arguments against transcendental idealism. The second direct argument is given by both Moore and Russell. Moore presents it in Principia Ethica, where he discusses an assump-

99 Russell attributes to Kant a view that is very similar, if not identical, to C. D. Broad’s interpretation. According to Broad (1978, pp. 3-8), Kant’s strategy was to explain the synthetic apriori by introducing a new category of judgments, transcendentally apriori judgments. Unlike traditional apriori judgments, they are not judgments of intrinsically necessary propositions: “[i]f Kant is right, they are judgments of propositions which are necessary consequences of certain facts about the human mind; but these facts are contingent and so are their consequences” (id., p. 7). 214 Ch. 3 Russell on Kant tion which he calls the “‘Epistemological’ method of approaching Metaphysics” (1903a, §78). This principle says that “by considering what is implied in cognition – what is its ‘ideal’ – we may discover what properties the world must have, if it is to be true” (ibid.) He suggests that it is a essential part of Kant’s Copernican revolution, and dismisses it (and the revolution that is dependent upon it) as a result of rather elementary confusions:

But though I cannot distinguish what is true from what I think so, I always can distinguish what I mean by saying that it is true from what I mean by saying that I think so. For I understand the meaning of the supposition that what I think true may nevertheless be false. When, therefore, I assert that it is true I mean to assert something different from the fact that I think so. What I think, namely that something is true, is always quite distinct from the fact that I think it. The assertion that it is true does not even include the assertion that I think it so; although, of course, I do think a thing true, it is, as a matter of fact, also true that I do think it. This tautologous proposition that for a thing to be thought true it is necessary that it should be thought, is, however, commonly identified with the proposition that for a thing to be true it is necessary that it should be thought. A very little reflection should suffice to convince anyone that this identification is erroneous; and a very little more will shew that, if so, we must mean by “true” something which includes no reference to thinking or to any other psychical fact. It may be difficult to discover precisely what we mean – to hold the object in question before us, so as to compare it with other objects: but that we do mean something distinct and unique can no longer be a matter of doubt. That “to be true” means to be thought in a certain way is, therefore, certainly false. Yet this assertion plays the most essential part in Kant’s “Copernican revolution” of philosophy, and renders worthless the whole mass of modern literature, to which that revolution has given rise, and which is called Epistemology. Kant held that what was unified in a certain manner by the synthetic activity of thought was ipso facto true: that this was the very meaning of the word. Whereas it is plain that the only connection that can possibly hold between being true and being thought in a certain way, is that the latter should be a criterion or test of the former (1903a, §80). Ch. 3 Russell on Kant 215

Russell’s verdict on “Epistemology” or “Epistemological method” is essentially the same as Moore’s. The point emerges several times in the Leibniz-book, most notably in sections 111-113, where Russell discusses Leibniz’s “deduction of God from eternal truths” (1900, §113). What he has in mind is the argument that essences and possibilities, insofar as they are real, must have that reality founded in something and this something is God’s understanding (Leibniz’s argument is found, for example, in Monadology, §§43-45). Russell judges this argument unsound for the reason, among others, that it rests on a failure to distinguish between God’s knowledge and the truths which God knows. The principle underlying Leibniz’s proof, to wit, that the eternal truths would not subsist if there were no understanding to think about them, has been encouraged, according to Russell, “by Kant’s notion that à priori truths are in some way the work of the mind, and has been exalted by Hegelianism into a first principle” (1900, p. 181). Like Moore in his dismissal of Epistemology, Russell detects a simple fallacy behind the first principle; Leibniz’s deduction is just a theological form of a more general argument from the premise that nothing can be true without being known to the conclusion that there is a mind, in some appropriate sense, “from whose nature truths perpetually flow or emanate” (ibid.). This argument fails because of the evident fact that “truths are something different from the knowledge of them” (id., p. 180).100

100 Russell gives a new twist to his discussion of Leibniz’s argument (and makes it a good deal more interesting), when he claims that the real ultimate premise on which the argument rests is not the view that truths depend upon knowledge for their existence but the idea which he calls the existential theory of judgment. This is the theory that “all truth consists in describing what exists” (1900, p. 182). The dependence of truth on knowledge then becomes just a special case of the existential theory of judgment: “[f]or truth is evidently something, and must, on this theory, be connected with existence. Now knowledge (perhaps) exists, and therefore it is convenient to make truth a property of knowledge. Thus the proposition, that a given proposition is true, is reduced to the proposition that it is known, and thus becomes existential” (ibid.) The best way to make sense of the existential theory of 216 Ch. 3 Russell on Kant judgment (ETJ for short) and Russell’s analysis of the conceptual situation is to compare them with what Robert Adams (1994, Ch. 7) has written about Leibniz’s argument (I am grateful to Markku Roinila for drawing my attention to Adams’ work on Leibniz). According to Adams, Leibniz’s deduction is based on the thesis that “whatever is true (possible), there must be something in virtue of which it is true (or possible)” (id., p. 178). Russell and Ad- ams differ in that where the former speaks of truths as descriptions of what exists or of truths as being about what exists, the latter speaks about truths being grounded in or being true in virtue of something or other. It is natural to hold that the two notions, being about and being true in virtue of, are not unconnected. For surely there is some systematic connection between what a proposition is about and that in virtue of which it is true (see Smith (1999) for an attempt to connect the two). It must be said, though, that Adams’ formulation is superior to Russell’s. Firstly, Adams makes Leibniz’s starting- point more commonsensical or less esoteric. For Adams writes that Leib- niz’s presupposition has “great intuitive appeal and [is] by no means peculiarly Leibnizian” (ibid.) This seems to be quite correct. For Leibniz’s presupposition – the thesis that truths, be they ordinary, non-modal or modal truths, require something in virtue of which they are true; that they need “truth-makers” or “an ontological grounding” – has attracted a fair amount of attention recently. Although there is no consensus on its correct formulation, the intuition behind it is often taken to be fairly evident, perhaps even platitudinous. (Some care must be exercised here, though. There is no consensus on where platitudes end and controversial philosophical theorizing begins; for example, Julian Dodd (2000, p. 4, fn. 8) argues that the locution “in virtue of” (as in “there is something in the world in virtue of which a statement is true”) should be banned, because it’s precisely by using it that philosophers end up confusing the correspondence platitude with what is no longer “uncontroversially a part of our conception of truth” (ibid.), namely the idea that truths are made true by something in the world.) Russell’s use of ETJ, by contrast, tends to obscure the real character of Leibniz’s reasoning. As Russell uses it, ETJ is not the ontologically neutral or “innocuous” requirement that truths must be grounded, one way or other, in the nature of things; rather, it is the far from uncontroversial thesis that all truths have a grounding in the realm of the existents. This is shown by two considerations. Firstly, Russell is tempted to see in ETJ a reduction of truth to something existential: truth, as Russell puts it, consists in describing what exists. For example, Leibniz is said to have reduced truth to knowledge: “thus the proposition, that a given proposition is true, is reduced to the proposition Ch. 3 Russell on Kant 217

The third direct argument is found in the Principles, §430, where Rus- sell discusses certain arguments that Lotze had presented for the apriority of space. The discussion includes the following comment:

The fifth argument seems to be designed to prove the Kantian apriority of space. There are, it says, necessary propositions concerning space, which shows that the nature of space is not a “mere fact.” We are intended to infer that space is an à priori intuition, and a psychological reason is given why we cannot imagine holes in space. The impossibility of holes is apparently what is called a necessity of thought. [...] Concerning necessities of thought, the Kantian theory seems to lead to the curious result that whatever we cannot help believing must be false. What we cannot help believing, in this case, is something as to the nature of space, not as to the nature of our minds. The explanation offered is, that there that it is known” (1900, p. 182). But this kind of reduction was hardly on Leibniz’s agenda, and if we follow Adams’ interpretation, we are not even inclined to think so. Secondly, Russell argued that ETJ was a consequence of a neglect of “Being” (as opposed to existence); that an advocate of ETJ has failed to appreciate the fact that a thing that does not exist need not be nothing for that reason alone. Again, this is simply false, as applied to Leibniz. For Adams shows that Leibniz was in fact well aware of the chief alternatives to his own account: that eternal truths are grounded in, or true in virtue of, Platonic entities (a realm of Being, this being Russell’s option); that they are grounded in some features of human thought (the “anthropological” model propounded by Hobbes). Leibniz, then, did not neglect the first alternative; on the contrary, as Adams shows, he argued against it on the grounds that the objects of mathematics and logic are not the sort of entities that could subsist on their own (for details, see Adams 1994, pp. 179-180). Let it be noted in passing, that Moore, too, refers to ETJ – or at least something like it – as the ultimate premise of the Epistemological Method; see Moore (1903a, §73). Like Russell, Moore succeeds in engendering a good deal of confusion, because they are both inclined to interpret ETJ as a thesis about meaning (this is explicit in Moore): they are inclined to think, that is, that anyone who says that a truth about an entity is grounded in something existent is trying to offer an analysis of the meaning of propositions featuring that entity in terms of some other entities that are existent. Given their views on the meaning of propositions, this amounts to finding the real constituents of propositions which are about that entity. 218 Ch. 3 Russell on Kant

is no space outside our minds; whence it is to be inferred that our unavoidable beliefs about space are all mistaken.

I would suggest that we interpret the argument of this rather dense passage in the following way. Kant’s explanation of the apriori foun- ders on the fact that it makes all our ordinary beliefs about space and its properties false, because it involves a mistake about their truth- makers. Let x be a proposition of which we would ordinarily say that it is true and is about space and its properties; yet, if x is the sort of proposition that Kant argues is apriori, we would have to conclude, if we give Kant’s doctrine a rather Russellian paraphrase, that x is also supposed to be true in virtue of our belief. Now, the most natural reading of this “in virtue of” -locution is that x is made true by our belief or by the constitution of our minds, and hence that x is really about our minds and not about the character of space.101 And yet, anyone who endorses x believes something about space and its properties and not about the constitution of the mind. Hence, anyone who accepts x – and by Kant’s theory, that is everyone, since we cannot but believe x – endorses a proposition that is false: after all, there is no space to make x true; and if it is retorted that the correct formulation is that “there is no space outside our minds”, this will do little to help Kant’s cause, because there is nothing in the proposition x to suggest that it is about (or is made true by) the constitution of our minds and not the constitution of space. Russell’s argument could be explicated as follows, inserting comments in square brackets:

What we cannot help believing, in this case, is something as to the nature of space, not as to the nature of our minds. [What we cannot help believing is the proposition that there are no holes in space; this proposition is about space and not about our minds.]

101 And not only is it the most natural reading; clearly, the status which Kant gives to space requires that at least some propositions about it are true. Ch. 3 Russell on Kant 219

The explanation offered is, that there is no space outside our minds; [What needs to be accounted for is why, although the explanation of why there can be no holes in space is that we cannot imagine them, the proposition that there cannot be holes in space is nevertheless about the nature of space and not about our minds; and the explanation is that facts about space are (identical with) facts about the constitution of our minds.]

whence it is to be inferred that our unavoidable beliefs about space are all mistaken. [Because there is nothing in the proposition “there cannot be holes in space” to suggest that it is concerned with our minds rather than the nature of space.]

According to this reconstruction, transcendental idealism about space amounts to the following, somewhat complicated view. Our beliefs about the nature of space involve a systematic mistake about the truth-makers of these beliefs. Insofar as we construe them as beliefs about space (and that is how we must construe them, if we are to believe Kant), they are all false, because they are, in fact, not about the nature of space but about the constitution of our mind. This reconstruction of Russell’s argument is rather speculative, and reading §430 of the Principles this way, I may be reading too much (or too complicated an argument) into it. And yet the above argument appears to be a perfectly intelligible way to spell out the difficulties that one may have in trying to grasp the essential content of transcendental idealism. There is one question, though, which the interpretation does not address: How is our mind (or the mind, or some such thing) supposed to decide the truth-values of our assertions about space and its properties? What, in other words, could it mean to say that beliefs that we are bound to construe as being about space are really about our minds? The suggested reconstruction bypasses this question and focuses on a consequence which it claims to be able to derive from Kant’s doctrine, namely, that all our assertions about space are false. Russell’s argument could, however, be given a reading that centres, precisely, on this question; in that case his contention 220 Ch. 3 Russell on Kant would be that Kant’s theory fails to make sense of the crucial notion “true in virtue of our beliefs” (or “true in virtue of the constitution of our minds”, etc.). Understood in this way, Russell is seen as address- ing an issue we saw Moore raising against transcendental idealism in his Kant’s Idealism. According to this line of thought, the claim that a proposition which is about the properties of space is “true in virtue of” our belief, if it means anything intelligible, can only mean that our mind is so constituted that it makes us think in a certain way and that way is spatial. Both of these interpretations make transcendental idealism an error theory in Mackie’s sense; our unavoidable beliefs about the nature of space are all false, because they suggest that there is space outside our mind, when there really is no space outside our minds. Whether we diagnose the error in the first way – our beliefs about the nature of space are really about something else, namely, our minds – or in the second way – our beliefs about the nature of space have no other source than a psychological mechanism which imposes such beliefs upon us – the conclusion is the same in both cases: we think and talk “as if p, when in fact p is false”.102 How should we assess these interpretations and the arguments to which they give rise? We should note, to begin with, that, as a rule, Russell’s evaluation of the prospects of Kant’s theory is predicated on the second interpretation. Since an argument which presupposes this interpretation is no longer direct in our sense – such an argument admits that transcendental idealism is a comprehensible doctrine but purports to show that it has consequences which are plainly unac- ceptable – further discussion of it is best deferred until we come to consider Russell’s indirect arguments. We are thus left with the first interpretation. Consider, then, the claim that transcendental idealism implies that there is a class of special propositions which are true solely in virtue of being believed; or, paraphrasing this in a manner that makes it

102 I borrow this as if -formulation from Blackburn (1986, p. 122). The context of Blackburn’s discussion, however, is different. Ch. 3 Russell on Kant 221 more intelligible, that the propositions in this class – among these a number of propositions about the nature of space – are true in virtue of the constitution of our mind. I think it must be said, as against Russell, that it is highly unlikely that transcendental idealism, as Kant conceived of it, is committed to any such straightforward thesis about the truth-makers for these propositions. The details of this exegetical issue need not be pursued here. Supposing, however, that this really was Kant’s view, we ought to ask why it should be thought to imply a commitment to error theory. It seems altogether more reasonable to think that Kant endorsed a “non-standard” view of the truth-makers of these special propositions: propositions about the nature of space are made true by facts about the constitution of our minds, because facts about space are facts about our minds and its constitution. Stated in this simple fashion, Kant’s theory may strike us as wildly implausible; how could facts about space really be identical with facts about the mind in any sense of the latter term that we can honestly claim to understand? But the idea itself that facts of a certain kind, rather than being accepted as they are, are best identified by facts of some other kind is commonplace in philosophy; after all, philosophers have always been in the habit of proposing extraordinary truth- makers for ordinary truths (contemporary analytic philosophers are no exception in this respect, witness the discussion that David Lewis’ “genuine modal realism” has inspired). If we ask, then, how Russell intended to distinguish between those would-be identifications (claims to the effect that entities of a kind K consist of entities of some other kind, L) which are like Kant’s in that they can be dismissed by the simple error-theoretic strategy and those that cannot be dispensed with so lightly, the answer seems to be that the argument which I tentatively attributed to Russell offers no means to draw a distinction between the two cases. That is to say, the reconstructed argument of §430 does not in fact depend upon the specifics of the Kantian theory of space. Any proposed identification of facts of kind K (facts about the nature of space) with facts of kind L (facts about the constitution of our minds) can be met with the Russellian strategy, i.e., by claiming that propositions about Ks (like 222 Ch. 3 Russell on Kant the proposition that there can be no holes in space) are about Ks (about space) and not about “something else” (the constitution of our minds). And if this conclusion is correct, perhaps it supplies a good reason not to consider the first interpretation of Russell’s argument any further.103

3.6.3.3 Three indirect Arguments against the R-model

3.6.3.3.1 The Consequences of the R-model

Even if these direct arguments are set aside, there is still plenty to complain about the r-model and its alleged explanatory capacity: it is to these that I shall now turn, as they form the real core of Russell’s case against Kant. Russell has more than one indirect argument against the r-model. These arguments do not question the model on grounds of intelligibility or some other such property. They try to show instead that though intelligible, the model has consequences that cannot be accepted. Though several in number, all these indirect arguments are in fact variations on a common theme. In the simplest possible terms,

103 This type of argument (if such it be) is usually associated with Moore, rather than Russell. It is characteristic of Moore’s defence of common sense (Moore 1925), but surfaces occasionally in his earlier writings, too. The following quotation from is a case in point: “[i]f to say that matter exists is simply equivalent to saying that the categories do apply to it, he [sc. Kant] does hold that matter exists. But the fact is that the two statements are not equivalent: I can see quite plainly that when I think that chair exists, what I think is not that certain sensations of mine are connected by the categories. What I do think is that certain objects of sensations do really exist in a real space and really are causes and effects of other things. Whether what I think is true is another question: what is certain is that if we ask whether matter exists, we are asking this question; we are not asking whether certain sensations of ours are connected by the categories” (Moore 1903-04, p. 140; emphases in the original). Ch. 3 Russell on Kant 223 the claim is that Kant’s explanation of the apriori cannot be correct, because the putative transcendental foundation – the constitution of our mind – is too weak to sustain the properties that characterize the apriori. There is therefore an irreconcilable mismatch between what a description of this constitution can deliver in principle and what is required of a proper explication of the notion of apriori knowledge (or of apriori proposition). From this schematic argument we can derive a number of instances by considering each of the attributes that together form the content of the concept apriori knowledge of apriori proposition. I shall assume that the r-model takes the following form. The goal is to explain why a synthetic proposition, p, is apriori, and the explanation consists in pointing out a feature, F, that belongs to p and is related to our cognition (expresses a property of our mind, as Russell might have put it). Thus the explanation itself is of the form

(1) Apriori(p), because F(p),

According to traditional accounts of apriori knowledge, a proposition that is apriori has three interesting properties: (1) it must be true;104 (2) it must be necessary (or necessarily true); (3) it must be universal.105 Hence, (1) divides into

(3) p is true, because F(p) (4) p is necessary/necessarily true, because F(p)

104 Of course, truth is not the privilege enjoyed only by (some) apriori propositions. It is nevertheless reasonable to include it among the consequences of apriority, because the explanation of why they are true reflects their apriority. 105 “Apriori” applies primarily to cognitions: what is apriori in the primary sense is not a truth but our knowledge of it (Stenius 1981, p. 346). But nothing stands in the way of extending its use to the truths themselves: a truth (a proposition) is apriori if it is knowable independently of sense- experience. According to philosophical tradition, truths that are thus knowable are necessary and (perhaps) universal. 224 Ch. 3 Russell on Kant

(5) p is strictly universal, because F(p)

It follows, then, from the r-model that p’s possessing some cognition- related feature, F, must explain why p is true, why it is necessary (or necessarily true) and why it is universal (universally valid). Accord- ingly, insofar one is dissatisfied with (1), there arises the possibility of raising three distinct, though not unrelated, objections to it: firstly, one may object that (1) fails to explain why p is true; secondly, one may object that (1) fails to explain why p is necessary; thirdly, one may object that (1) fails to explain why p is universally valid. As we shall see, Russell endorses each of the three arguments. In what follows I shall start by considering Russell’s objection to (4), which I shall call the Argument from Necessity. I shall then turn to (3) and Rus- sell’s argument against it, which I shall call the Argument from Truth. Finally, I shall consider (5) and the Argument from Universality that Rus- sell uses to undermine it.

3.6.3.3.2 An Argument from Necessity

In general, Kant explains a range of properties that synthetic apriori propositions have by tracing the property-possession to our cognition of those propositions. Considering the Argument from Necessity, the following formulation suggests itself. The r-model, it would seem, starts from some fact about “our nature”, or about our “human cognitive capabilities”, as James van Cleve puts it in the course of considering the argument (van Cleve 1999, p. 37). It is a natural assumption that what cognitive capabilities human beings have is a contingent matter (ibid.) From this apparently contingent starting-point, the r- model claims to derive a conclusion that holds necessarily. This sets the stage for an obvious objection, which Russell formulates on more than one occasion. van Cleve cites the following passage from the Problems of Philosophy: Ch. 3 Russell on Kant 225

The thing to be accounted for is our certainty that the facts must always conform to logic and arithmetic. To say that logic and arithmetic are contributed by us does not account for this. Our nature is as much a fact of the existing world as anything, and there can be no certainty that it will remain constant. It might happen, if Kant is right, that to-morrow our nature would so change as to make two and two become five. This possibility seems never to have occurred to him, yet it is one that utterly destroys the certainty and universality which he is so anxious to vindicate for arithmetical propositions (1912a, p. 49; emphasis added).

The same point is also made in the following passage, which is from Russell’s review of Poincaré’s Science and Hypothesis:

The notion that a principle is rendered certain by expressing a property of the mind is also curious. “The mind” must be somebody’s mind; all minds are a part of nature; minds differ from time to time and from person to person; and psychology is not usually considered more certain than arithmetic. M. Poincaré’s view, like Kant’s, assumes that we know already, before we have any other knowledge, that all minds are alike in certain respects; that their likeness consists in their sharing certain beliefs; that these beliefs have no warrant except their universal existence, i.e. that they are universal delusion; and that universal delusions are what we call à priori truths, and as such are the indispensable premisses of all really indubitable knowledge. (1905b, p. 591; emphasis added)

(There is more to this passage than just a point about necessity; what matters now is the italicized passage) In both of these passages Rus- sell makes his point in terms of certainty and not necessity. However, elsewhere in the Problems of Philosophy he speaks freely about necessity and not just certainty. Although Kant is not mentioned in the following passage, the problem it is about is precisely that of apriori knowledge, and one of the points he is making is that we are accustomed to thinking that apriori propositions are more than just well-entrenched empirical generalizations:

[W]e feel some quality of necessity about the proposition ‘two and two are four’, which is absent from even the best attested empirical generalizations. Such generalizations always remain mere facts: we feel that there 226 Ch. 3 Russell on Kant

might be a world in which they are false, though in the actual world they happen to be true. In any possible world, on the contrary, we feel that two and two would be four: this is not a mere fact, but a necessity to which everything actual and possible must conform. (1912a, p.43)

Furthermore, there is a passage in the Principles of Mathematics in which the argument from necessity is explicitly directed against a Kantian account of necessity. The relevant parts of the passage run as follows:

The fifth argument [an argument by Lotze against points] seems to be designed to prove the Kantian apriority of space. There are, it says, necessary propositions concerning space, which show that the nature of space is not a “mere fact.” We are intended to infer that space is an à priori intuition, and a psychological reason is given why we cannot imagine holes in space. The impossibility of holes is apparently what is called a necessity of thought. [...] we only push one stage further back the region of “mere fact,” for the constitution of our minds remains still a mere fact. (1903a, §430)

In fact, Russell gave this argument as early as 1898, during his transitional or Moorean period, in a paper entitled “Are Euclid’s Axioms empirical?” In the course of discussing the criteria of the apriori, he claims that “[w]e must disregard entirely the empirical fact of our knowledge and consider only what is necessary” (1898b, p. 333). He then argues, to emphasize the point he has just made, that what he calls “our psychical nature” is entirely irrelevant in the study of the apriori. The reason for this the following: “Our psychical nature, except to the extent that it is bound by the à priori laws governing everything that exists, seems to be entirely empirical; this is a given fact, not a necessary truth” (ibid.) What is the argument of these passages? van Cleve (1999, p. 38) suggests that Russell had the following argument in mind. Kant thought that the necessity that judgments that are synthetic and apriori, like mathematical judgments, owe their necessity to our “cognitive constitution”. But it is natural to think that this constitution is a contingent matter: “our nature could change, or it might have been Ch. 3 Russell on Kant 227 different originally even if for some reason it cannot change” (ibid.) If this is admitted, it follows that the laws of, say, arithmetic and geometry are not necessary after all; if our constitution had been different, either originally or as a result of a sufficiently radical change, the laws would have been different. This, however, is absurd. Hence, Kant’s explanation of why such laws are necessary cannot be right.106 This is surely a very natural reading of the passages just quoted (I will point out later, however, that Russell’s argument is slightly more complicated). Van Cleve continues by considering a possible rejoinder to Russell. Here the leading idea is that even though Kant’s thesis is that the ground of synthetic apriori truths is ultimately factual, he is not denying the necessity of mathematics; he is only denying that mathematical laws are necessarily necessary. That is, he is only denying the characteristic axiom of the modal system S4: ɷp Ⱥɷɷp. According to this defence of Kant, what emerges from Russell’s discussion is the following putatively perfectly harmless argument (id. pp. 38-39). We start from the characteristically Kantian premise that there is an important class of propositions – like the propositions of geometry and arithmetic – which owe their necessity to a feature that is related to our cognition. This premise van Cleve symbolizes as follows:

1. (p)(ɷp Ⱥ [ɷp Fp]

106 van Cleve says, regarding the quotation from the Problems of Philosophy, that Russell suggests that, according to Kant, there is a connection between our forms of intuition and the truths of logic (1999, p. 38). In fact, however, Russell suggests no such thing. What he says is that, according to Kant, logic and arithmetic are “contributed by us”, or that they have something to do with “our nature”. Since Kant thought that logic – general or formal logic – has something to do with reason, and since reason is presumably a part of “our nature”, Russell’s remark is perfectly correct. It must be admitted, though, that in the Kantian context the necessity of formal logic calls for a treatment quite different from that of arithmetic. 228 Ch. 3 Russell on Kant

Here “ɷ” stands for necessity, “Ⱥ” for material implication and “” for strict implication or entailment. What 1. says, then, is that if a synthetic proposition is necessary, it is so because it is grounded in “our nature”.107 Van Cleve calls 1. the Dependency Premise. Next we add the premise that it is contingent and therefore possibly false that any given proposition is delivered by the form of our intuition:

2. (p) ¹¬Fp

This is called the Contingency Premise. From 1. and 2. we continue as follows. Let p be any synthetic and necessary proposition: we have:

3. ɷp

1. and 3. together yield, by Modus Ponens, the following:

4. ɷp Fp

It is an incontrovertible principle of modal reasoning that if one proposition entails the other, and the second is possibly false, so is the first. Thus, from 2. and 4. we infer:

5. ¹¬ ɷp

Finally, 5. is equivalent with

6. ¬ ɷɷp.

107 van Cleve discusses only the case of geometry. Hence, in his presentation the quantifier in 1. ranges over all geometrical propositions and “Fp” says that p is a “deliverance of our intuition” (id., p. 38). I shall speak, more generally, about all synthetic propositions that are necessary; hence ‘Fp’ says something to the effect that p is delivered by or grounded in our cognitive capabilities; such finer details as these make no difference to the argument itself, however. Ch. 3 Russell on Kant 229

If this argument is accepted, we must conclude that the characteristically Kantian Dependency Premise does not yield the result that a proposition which is delivered by the form of our intuition is not necessary; the only conclusion is that its modal status (in this case, its necessity) is itself a contingent matter. And, as van Cleve puts it, “this is a conclusion that some people are quite prepared to live with” (1999, p. 39). On the other hand, the notion of contingently necessary is often dismissed as absurd. What Russell would have thought about it we do not know, since he never discussed such modal subtleties.108 How- ever, van Cleve argues that we need not in fact debate the merits of S4 here. For he provides a slightly modified version of the above argument, which, if sound, shows that a Kantian cannot, after all, maintain necessity for what is delivered by the form of our intuition. This modified argument starts with the following strengthened version of the Dependency Premise:

1.´ (p)(ɷp Ⱥ [p Fp])

What 1.´ says is that if a proposition owes its necessity to the character of our mind, the form of our intuition, then it owes its truth, too, to that form. This assumption, it must be admitted, sounds eminently plausible.109

108 But see Dejnozka (1999). 109 Suppose, for example, that the F in 1. and 1.´ denotes not a feature of human cognition but, rather, the will of God. The original Dependency Premise would then say that a proposition is necessary, because God has decided so, a view that Descartes accepted with his doctrine of the creation of eternal truths (van Cleve 1999, p. 40). But if it is thought that p’s necessity is due to God’s will, it would be odd to deny this for its truth: “[t]o believe otherwise is to attribute to God a queer form of omnipotence that holds sway over truths of the form ‘ɷ’ but not over truths generally” (ibid.) In the same vein, it would be a strange form of Kantianism if one held that the propositions of geometry owe their necessity to the form of our intuition, but that their truth is entirely independent of it. 230 Ch. 3 Russell on Kant

With 1.´ granted, the modified argument goes as follows (van Cleve 1999, p. 40):

1.´ (p)(ɷp Ⱥ [p Fp]) The Strengthened Depend- ency Premise 2. (p)¹¬ Fp Contingency Premise 3. ɷp Assumption for reductio 4. p Fp from 1. and 3. by Modus Po- nens 5. ¹¬ p from 4. and 2. by modal logic 6. ¬ ɷp from 5. by the interdefinabil- ity of ¹ and ɷ

In other words, if the Strengthened Dependency Premise is accepted – and it makes little sense to deny this if the original Dependency Premise is accepted – it must be concluded that a proposition which owes its necessity to an ultimately contingent factor like our nature or our cognitive capabilities loses its necessity. But this seems like bad news for the Kantian: one can retain Kant’s explanatory strategy only if one is prepared to live with the possibility that a sufficiently radical change in our nature could bring about a change in the laws of, say arithmetic; if Kant is right, it might happen, as Russell puts it in one of the quotations given above, that to-morrow our nature would so change as to make two and two become five. There are reasons to think, however, that van Cleve’s modified argument does not in fact settle the issue. A defence of Kant on this issue would start from the observation that necessity and possibility, as they are used in the Kantian context and hence, also, in the previous pair of arguments, are crucially ambiguous; the claim is, in other words, that van Cleve has failed to pinpoint the sense in which such synthetic necessities as the propositions of arithmetic and geometry are necessary for Kant. In particular, insofar as we are followers of Kant’s modal theory, we must distinguish between logical and real necessity and possibility. The former rests on the law of contradiction, whereas the latter is delineated by the conditions on possible experi- Ch. 3 Russell on Kant 231 ence (categories and forms of intuition). Armed with this distinction, we can go on to argue that a proposition which is logically contingent may nevertheless be “really necessary”, and therefore that even if a mathematical proposition is logically contingent, it need not lose its necessity for that reason alone. How is this idea best cashed out? As in all discussions that relate to modalities, the first thing that comes to mind is the possible worlds idiom. On this understanding of modalities, it is all but inevitable to start from the notion of logical possibility, or the idea that a proposition is possible if it does not yield a contradiction by itself, and say that this notion delineates the widest domain of possibility, the domain which includes “all logically possible worlds”. From this we move on to real possibility by adding further constraints on the original set of worlds, that is, by limiting the original, or widest, domain of possibility. Given the possible worlds idiom, the appropriate correlate of Kant’s notion of “real possibility” would be the notion of a “really possible world”; this is suggested by Gordon Brittan (1978, pp. 21- 23), who uses world-talk to throw light on Kant’s concept of real possibility. A world that is “really possible”, Brittan explains, is a world that “beings like ourselves, endowed with certain perceptual capacities and conceptual abilities, could experience” (id., p. 21).110 What we could and could not experience is of course determined in the familiar Kantian manner: “a really possible world is a world whose limits and general form are given by the Categories, a world having a particular spatial-temporal-causal form that contains endur- ing centers of attractive and repulsive forces” (ibid.) Understood in this way, the notions of real necessity and real possibility qualify as merely relative modalities.111 The idea is that often, when a proposition is said to be necessary, all that is meant is that it is necessary relative to some chosen set of (true) propositions. These would be, very roughly, the “laws” that fix the necessity in question. Simi- larly, a proposition’s possibility is relative, when it is compatible with

110 See Patricia Kitcher (1990, p. 15) for a similar interpretative strategy. 111 For these notions, see Hale (1996), (2002, pp. 280-282). 232 Ch. 3 Russell on Kant the propositions in the privileged set. Standard examples of relative modalities are physical necessity and physical possibility: a proposition is physically necessary if it is a logical consequence of the laws of physics, and it is physically possible if it is compatible with those laws. Furthermore, and this is what makes the notion of relative modality an interesting one, a kind of necessity is merely relative, when the relevant propositions – or laws which delineate the necessity in question – are themselves contingent; that is to say, a kind of necessity, ƶ- necessity, qualifies as merely relative, when there is available another kind of necessity, Ƹ-necessity, so that there is a proposition, p, such that it is ƶ-ly necessary that p but Ƹ-possible that not-p (Hale 2002, p. 282-283).112 For example, it is a standard assumption that propositions which are physically necessary in the sense that they follow logically from the laws of nature are necessary only in this relative way. For there are senses of possibility – including, at least, logical and metaphysical possibility – such that a proposition that is physically necessary could nevertheless have been false in these other senses. Returning to the Kantian context, we see that real necessity and real possibility, on their current understanding that draws on world- talk, qualify as merely relative modalities. A proposition, p, is really necessary, because there is a body of propositions which together express the conditions on possible experience and of which p is a logical consequence. But real necessity is merely relative, because the conditions on possible experience are themselves contingent. How do these considerations relate to van Cleve’s arguments? To begin with, recall the Dependency Premise in its original formulation:

1. (p)(ɷp Ⱥ [ɷp Fp]

112 As Hale (2002, p. 281) explains, this additional characterization of a modality as merely relative is needed, because every necessity qualifies trivially as relative, given our definition of relative necessity; “in particular, since the truths of logic are logical consequences of the empty set of premises, they will be logical consequences of any collection of any set of true propositions whatever” (ibid.) Ch. 3 Russell on Kant 233

Since we seek to defend Kant, we must “go Kantian” in our elucidations of modalities. That is, we musk ask, what kind of necessity are we dealing with here? F is supposed to capture those conditions that together characterize the notion of possible experience. The necessity in question is therefore relative to this set of conditions; hence, the sort of necessity that is relevant for the Dependency Premise is real necessity. 1. should therefore be represented as follows (using ‘ɷR’ for real necessity):

1.´´ (p)(ɷR p Ⱥ [ɷR p Fp]

Consider, next, the Contingency Thesis. What kind of modality pertains to it? A proposition that is delivered by the form of our intuition is necessary in the relative sense, because it is a logical consequence of propositions characterizing this form. But the Contingency Thesis concerns the modal status of this form itself, saying that it could have been something else. Hence the possibility that features in 2. must be logical. 2. should therefore be represented as follows (letting ‘¹L’ stand for ‘it is logically possible that’):

2.´´ (p)¹L ¬ Fp

This may be called the Logical Contingency Premise. Next we assume that p is necessary. Obviously, in the Kantian context the intended meaning is “really necessary”: p is delivered by the form of our intuition and qualifies therefore as really necessary. Our assumption, the Real Necessity Premise, thus receives the following form:

3.´´ ɷR p

From 1.´´ and 3.´´ we now infer:

4.´´ [ɷR p Fp] 234 Ch. 3 Russell on Kant

And from 2.´´ and 4.´´ we derive:

5.´´ ¹L ¬ ɷR p

5.´´ says that it is logically possible that p is not really necessary, or equivalently, that it is not logically necessary that p is really necessary:

6.´´ ¬ ɷLɷR p

But this is precisely as it should be, given our current understanding of Kant’s views on modality. All we have established so far is the characteristically Kantian distinction between logical and real necessity, together with their interrelations: real necessity is merely relative, because there is a sense of possibility – logical possibility – in which what qualifies as necessary in the relative sense could have been false. Let us now turn to van Cleve’s second argument. If sound, this argument shows that “there is no necessity” (van Cleve 1999, p. 39). We start with the strengthened version of the Dependency Premise, which we disambiguate in the manner now familiar:

1.´´ (p)(ɷR p Ⱥ [p Fp]

That is, if p is really necessary, its truth is delivered by the form of our intuition. From this premise we proceed as follows:

1.´´´ (p)(ɷR p Ⱥ [p Fp] The Strengthened Depend- ency Premise 2.´´ (p)¹L ¬ Fp The Logical Contingency Premise 3.´´ ɷR p The Real Necessity Premise 4.´´´ p Fp from 1. and 3. by propositional logic 5.´´´ ¹L ¬ p from 4. and 2. by modal logic 6.´´´ ¬ ɷL p from 5. by the interdefinabil- ity of ¹ and ɷ Ch. 3 Russell on Kant 235

Running through this argument, we reach the conclusion that a proposition which is delivered or necessitated by our nature is not logically necessary. Again, however, it can be argued that this is entirely in keeping with Kant’s theory of modality, and should cause no concern for any Kantian; since, logically speaking, our form of intuition (and any other factor that contributes to the delineation of “possible experience”) could have been different, its consequences – propositions delivered by it – are, indeed, logically contingent. But this does not show that “there is no necessity”; any proposition that is licensed by the form of our intuition is necessary in the sense introduced by Kant: they possess real necessity. What are we to make of this conclusion? It must be admitted, I think, that even though it sees off van Cleve’s argument in its original form, it raises at least as many questions as it answers. In particular, one who is critical of Kant is likely to raise an objection to the ultimate conclusion of the above argument. This objection grants that the distinction between logical and real modalities succeeds in retaining a vestige of necessity with the help of the doctrine that synthetic apriori propositions are necessary in the merely relative way. This, however, does little by way of rescuing Kant, because there are cases of necessity where merely relative necessity is not enough. For among the different kinds of propositions which Kant classifies as synthetic and apriori there are at least some whose necessity is not relative in the sense explained above but must be regarded as absolute. That is, there are kinds of propositions in the relevant class which are such that there is no sense of “possible” in which they could have been false. It is probably not a coincidence that Russell, when he discussed Kant’s explanation of the apriori in the Problems of Philosophy, chose arithmetic and logic as examples. For there is a very strong intuition among philosophers – not shared by everyone but nevertheless very common – that logic and arithmetic are cases in which it would be palpably inappropriate to think that the necessity which qualifies them is merely relative. And even if we put logic aside on the grounds that in the Kantian context logic – in the sense of formal or general 236 Ch. 3 Russell on Kant logic – calls for a treatment that is entirely different from what is given to arithmetic, the case of arithmetic nevertheless remains. It seems to follow from Kant’s account of arithmetic that its judgments are grounded – as regards their content – in time as the form of our outer intuition. If, then, there is a genuine possibility that our intuition could be different from what it in fact is (either because it was so originally or because it could change into something different), this possibility disqualifies that form from acting as the metaphysical source of arithmetical necessity. If it is granted that our nature is, as Russell puts it, “a fact of the existing world”, the conclusion must be drawn that this nature is an unsuitable candidate for the task of grounding any necessity that should be taken as absolute, rather than merely relative. Is there anything that can be said in defence of Kant? Since we are at present considering Kant’s views on necessity and possibility, I shall not now consider the other way in which the above argument might be sidestepped – namely, by denying that Kant ever held the view that arithmetical (and other synthetic apriori) judgments are necessary in the absolute sense; although Kant’s stand on the issue is not transparent, there is strong evidence that he did wish to maintain that these judgments are necessary in a stronger than the merely relative way.113 Considering, then, the modal question, it must be said, I think, that the prospects for a Kantian are distinctly bleak as long as he or she construes real possibility and real necessity as species of relative modality. The difficulty with Kant’s theory of modality, I would like to suggest, does not lie in the distinction between kinds of modality; the real root of the problem is the interpretation that we have imposed on the notion of real necessity. When we translated Kant’s distinction between logical and real modalities into world-talk, we started with the idea that logical possibility is the notion which delineates the widest domain of what is possible; that, in other words, logical possibility is the weakest sense in which something can be possible. Given this starting-point, further

113 See Brook (1992) for some discussion of this question. Ch. 3 Russell on Kant 237 and stronger senses of “possible” can be introduced by imposing limitations on the original domain. When such more restrictive senses are introduced, we single out particular regions from the entire logical space, or the set of all possible worlds. We do so, for example, when we propose to consider only those worlds which are not only logically possible but also comply with the laws of nature (the first constraint, logical possibility, is usually left unsaid, because we assume that our worlds are logically possible, whatever else they may be).114

114 One might suggest that, strictly speaking, the possible worlds interpretation of Kant does not presuppose that logical necessity is the strongest kind of necessity; that it only presupposes that logical necessity is stronger than real necessity (that “it is logically necessary that p” always entails “it is really necessary that p” but not vice versa). To this suggestion there are two replies. In the first place, in the Kantian context it is natural, to say the least, to make the following two assumptions: firstly, that logical necessity is absolute (at least as strong as any other kind of necessity), and secondly, that logical necessity is in fact the strongest kind of necessity (strictly stronger than any other kind of necessity). For there are, in this context, no candidates for a kind of necessity that would be stronger than logical (indeed, there are very few, if any, such contexts!). Nor are there kinds of necessity that would be serious candidates, in the Kantian context, for being as strong as logical necessity, i.e., kinds of necessity which are extensionally equivalent with, but distinct in content from, logical necessity. Secondly, we should note that the very definition of “relative necessity” presupposes that there is available a notion of absolute necessity. A proposition q is necessary relative to another proposition, p, if and only if q follows from p in the sense that excludes the possibility of p’s being true and q’s being false; if this exclusion were not absolute, we would not have a well-defined notion of consequence which we could use in the definition of relative necessity. This general point about the notion of relative necessity has been made by Bob Hale (1994, p. 317, fn 29. Hale uses the observation to argue that recognition of modality goes hand in hand with recognition of absolute modality, and that, therefore, the only real option for one who is disinclined to operate with a notion of absolute modality is the rejection of modality tout court (“logical irrealism”, as it is sometimes called)). Since we are accustomed to thinking that consequence is a matter of logic, we tend to think, more or less automatically, that the strongest sense of necessity available is logical. Hence, it turns out that the world- talk interpretation of Kant does presuppose that there is available an abso- 238 Ch. 3 Russell on Kant

Corresponding to this notion of logical possibility – logical possibility as the weakest kind of possibility – there is the notion of logical necessity as the strongest kind of necessity. If logical necessity is like this, then whatever other kinds of necessity there may be, these are merely relative (for every kind other kind of modality, ƶ, “it is logically necessary that p” always entails “it is ƶ-ly necessary that p”, but not vice versa). This so-called bi-polar picture of modality115 yields the following characterization of real possibility:

(i) there are propositions which are really necessary, but logically contingent; (ii) there are propositions which are logically possible, but really impossible.

That is to say, logical necessity is a genuinely stronger modality than real necessity, and real possibility is genuinely stronger modality than logical possibility. Given this interpretation of the logical vs. real distinction, there is no choice but to grant that Kant’s synthetic apriori has a basis that is genuinely contingent, even if this contingency should be only of the logical sort. The present interpretation makes the basis genuinely contingent, because the underlying bi-polar picture of logical necessity and possibility starts from a positive characterization of these modalities. That is to say, what is really possible results from a limitation on an antecedently given domain, or domain that is given independently of what is really possible. And if we understand the notion of real possi- lute notion of necessity, which is used, possibly among other things, to define the idea relative modality. Since this definition is framed in terms of “consequence”, it is natural to assume that the necessity in question is logical. This, incidentally, constitutes a rather grave objection to considering Kant’s real necessity as a species of relative modality; for there are good reasons to think that the notion of formal logic that was available to Kant is far too weak to sustain its use in a fully general definition of consequence, which could be put to use in a definition of relative necessity. 115 Cf. Hale (1996). Ch. 3 Russell on Kant 239 bility this way, it appears to be perfectly sensible to ask, counterfactu- ally, what would be the case, if these limitations were lifted. For us, this way of thinking about modalities is very natural, because we are accustomed to thinking about possibilities model- theoretically. That is, we can readily make sense of a possibility if we can spell it out with the help of the notion of a model, and models themselves can have as many – or as few – restrictions built into their characterization as we please. In Kant’s case, however, the situation is quite different for the simple reason that he had no comparable conceptual tools available to himself. And this point, familiar by now, goes into the very heart of the present issue: that is to say, there are good reasons to think that the bi-polar picture and the concomitant idea of relative modalities fails to capture the essence of Kant’s distinction between logical and real modalities. In what follows, I shall discuss these issues as they arise in the case of geometry. Kant’s theory of geometry involves, among other things, the following three claims. Firstly, geometry is the science which determines the properties of space qua form of our outer intuition. Secondly, geometrical judgments are necessary. Thirdly, this necessity is a consequence of the special role that (the representation of) space occupies in our experience. It is natural to regard the last of these claims as involving a commitment to the view that geometry, insofar as it is necessary, is so only in a merely relative way. To see how natural such a reading is, consider what Gottfried Martin (1955) has to say about Kant and geometry. Martin gives the following (partial) gloss on Kant’s claim that Euclidean geometry is based on intuition: “[i]ntuition [...] is not an additional source of knowledge for mathematics [...] but is the factor which limits the broader region of logical existence, namely what is thinkable without contradiction, to the narrower region of mathematical existence, namely what can be constructed” (id., p. 25; emphasis added). Here the ideas that we have been explor- ing are made almost explicit by an eminent Kant-scholar. “Logical existence”, or possibility qua absence of contradiction is, according to Martin, a notion that is broader than “mathematical existence”, i.e., what can be constructed, or is constructible. It is true that he does not 240 Ch. 3 Russell on Kant say that “logical existence” singles out the widest domain of possibility or that it is the weakest sense in which something is possible. He says only that it is “broader” – less restrictive – than mathematical existence. But I think we may safely suppose that this is Martin’s view. At any rate, as I have already suggested, the absolute character of logical modalities is at least implied in the Kantian context; there is no sense of possibility available in that context that would be a candidate for being more permissive than logical possibility. Natural though Martin’s interpretation is, it seems not to have been Kant’s considered view. The elements of an alternative interpretation can be seen by consider the following passage by Johann Schulze, a pupil of Kant’s:

According to Kant, the ground of the representation of space is merely subjective, and it lies, as he has proven, not in the limits of the power of representation, but merely in the innate special constitution of our capacity for intuition (Ansschauungsfähigkeit). Now, to be sure, it certainly cannot be proven that this space is absolutely necessary, i.e., that every being capable of thought has it and must represent those things which we call “outer” as things in space. For us, however, the representation which we have of space is given with such unconditional necessity and unalterability through the original constitution of our capacity for intuition, that it is for us absolutely impossible to think away space, or to think it in any other manner. Thus, if we wished to change even a single predicate which is known by us to belong to it, the entire representation of space would be abolished and become a non-entity (Unding). The absolute necessity of the connection of the predicate with the subject, and therefore the apodictic certainty of all geometrical postulates and axioms, is grounded immediately in the absolute necessity to represent space exactly as it is given to us through the original constitution of our capacity of intuition and not otherwise. (Whoever would deny this must give up the whole of space itself as the object of geometry). Moreover, this is also the mediate ground of the certainty of all those problems and theorems which can be derived merely from these postulates and axioms. If, on the other hand, that particular, original constitution of our capacity of intuition were given up, then space in its entirety would be nothing, and in that case all geometrical concepts and propositions would likewise be nothing. (Allison 1973, p. 172) Ch. 3 Russell on Kant 241

In this passage Schulze argues for – or at least states – the following four points. Firstly, he admits, like Kant, that space is not absolutely necessary in the sense that every thinking being would have to “use” space as a condition of individuation (cf. B72). Secondly, Space is nevertheless absolutely necessary for us. Thirdly, absolute necessity is shown by our inability to think space away or to think it as in any way different; if we changed as much as one predicate in our original representation of space, we would find ourselves no longer thinking about space; space itself, and thereby the object of geometry, would become a non-entity.116 Fourthly, our space representation, though absolute in this strong sense, is nevertheless grounded in something that can be characterized as a “mere fact”, to wit, the “original constitution of our capacity for intuition”. Schulze’s views are important, because they are formulated explicitly as a rejoinder to a Russell-style objection; the above quotation is written in reply to Eberhard, whose criticism of Kant’s theory of space concludes with the following statement: “[t]he imageable in space has only subjective grounds, namely, in the limits of the representing subject. This subjective element, however, is alterable and contingent. It is therefore impossible for it to be the sufficient ground of the absolute necessity of eternal truths” (Allison 1973, p. 172). The point that Eberhard makes in this quotation is precisely the same as Russell’s: our nature cannot ground absolute necessities or eternal truths, because it is subjective and, for that very reason, contingent and alterable. To this Schulze, the Kantian, replies by outlining a conception of necessity that is at least intended

116 Note how Schulze first explains “absolute necessity” as a conception that every thinking being has (must have?), and then moves to a characterization that is closer to our previous one: space, as it is “originally” given to us, is absolutely necessary, because we cannot make any sense of the idea that it might have been different. Thus, Schulze clearly thinks that the fundamental propositions characterizing space are absolutely necessary in the sense that there is no sense of possibility in which any of them could have been false; and this is so, because every attempt to frame such a “deviant” thought ends up with a confusion that can be only described as nonsense or, perhaps, as a “non-thought”; I shall return to this point below. 242 Ch. 3 Russell on Kant to be considerably stronger than any interpretation of the relevant kind of necessity as merely relative. Schulze’s defence of Kant offers only a sketch of this stronger notion, and a great deal more would have to be said to make it sufficiently precise, let alone attractive. But the main idea is reasonably clear. Schulze (and, we shall suppose, Kant) wants to give necessity a foundation that is genuinely transcendental, as opposed to one that is describable as merely factual (this latter idea is incorporated into the r- model). Schulze’s formulation of the idea is not as transparent as it ideally would be, however. For there are two distinct lines of thought that can be extracted from the above quotation, and Schulze, it seems, does not achieve a complete separation between them.117 The first line of argument is the more familiar one. It purports to show that something – we may call it simply a condition, keeping in mind, however, that there are different kinds of things that fall under this rubric – is necessary in the sense of being presupposed in our experience. And space is like this, because, among other things, we cannot think it away, as Schulze puts it.118 This line of thought, which focuses on the conditions of experience, is often identified as the key to Kant’s explanation of the possibility of synthetic apriori judgments. Russell and the r-model are no exception in this respect: according to the r-model, synthetic apriori

117 The following paragraphs draw heavily on Brook’s (1992) insightful discussion of Kant’s methods of recognizing necessary truths. I think it is only fair in this connection to bracket the fact that Schulze in the above passage, like Kant in the Transcendental Aesthetic, moves quite freely from “representation of space” to “space” and back, as if these were interchange- able terms. 118 Schulze’s formulation is presumably an allusion to the Transcendental Aesthetic, A24/B39: “[w]e can never represent to ourselves the absence of space, though we can quite well think it as empty of objects”. Kant takes this to show that (the representation of) space cannot be derived from experience but must be considered a “condition of the possibility of appearances” (ibid.) Ch. 3 Russell on Kant 243 judgments are consequences of certain very general facts about the constitution of the human mind, and these facts are, precisely, the conditions of possible experience. We have already seen, however, that reference to experience and its conditions is highly problematic, insofar as it is intended to safeguard the necessity simpliciter of a proposition or principle. And this, I have assumed, is the sort of conclusion that Kant really wanted to establish. The argument from the possibility of experience shows, if it shows anything, that there is a necessary connection between experience (of a certain kind) and a certain propositional or non-propositional condition; to use a different terminology, the argument shows, if it shows anything, that experience cannot be conceived without the condition in question, and that, therefore, we could not have experience in the absence of that condition – space, for example. If these are fair descriptions of Kant’s argumentative strategy, then his conclusions should be seen as propositions expressing conditional necessity:

(1) Necessarily, if we are to have experience of an object x, then C[...x...], where “C” expresses one of the conditions of experience. Clearly enough, (1) cannot be used to argue that C is necessary in any absolute sense: if there is a legitimate form of argument with the conclusion that C is a necessary condition for there to occur experience of x, that argument only shows that there can be no experience of x plus not-C. In other words, the argumentative strategy under consideration allows of the possibility that the connection between x and C fails to hold in those cases (“worlds”) in which there is no experience (and that, if the condition in question is propositional, its negation holds in such worlds). For this reason the fact – supposing it is a fact – that we cannot think space away, or that space is an inextricable element in our experience, shows at most that space is necessary in the presuppositional sense; and we have just seen that such presuppositions 244 Ch. 3 Russell on Kant themselves (propositions and principles which we must have in order to have experience) may be thoroughly contingent. Can the Kantian position be improved upon, if instead of experience of x, we focus on x itself, i.e., the object of experience? Such a move would seem to be licensed by Kant’s famous slogan, “[t]he a priori conditions of a possible experience in general are at the same time conditions of the possibility of objects of experience” (A111).119 If, then, the transition from ‘condition of experience’ to ‘condition of an object of experience’ is a legitimate one, would that not ensure that the sought-after connection, if it holds at all, holds between x and C, and does so necessarily? And if this is so, it would guarantee that the connection is of the right kind: if C is a condition that is necessary for there being objects of a certain kind, namely objects of experience, certainly those objects cannot exist, if C does not hold. And if this much is admitted, “necessarily, x is C” appears to be an absolute necessity: if C is necessary for the existence of x, there is no sense of possibility in which “x is not C” could have been true. The present suggestion is that Kant’s explanatory strategy, when correctly understood, yields results that are not of the form (1), but are, rather, as follows:

(2) Necessarily, if x is an object of experience, then x is C

I think, however, that this is best formulated slightly differently.120 The real Kantian strategy is not an argument to the effect that there is a set of conditions that are necessary for there to be experience (of a certain specified kind). And it is not even an argument that a number of conditions are necessary for there to be objects of experience. The

119 There is also the following, extremely revealing characterisation of transcendental proof by Kant; such a proof, we are told, “proceeds by showing that experience itself, and therefore the object of experience, would be impossible without a connection of this kind” (A783/B811; emphasis added), i.e., a connection which trades on the possibility of experience. 120 Again, I am indebted to Brook (1992). Ch. 3 Russell on Kant 245 real point is to argue that there can be no thoughts (of a specified kind) in the absence of certain conditions, or that we cannot conceive of x without a certain condition. To put the point another way, the Kant- ian strategy is to argue for the necessity of a condition on the grounds that it figures in our conception of what it is to be an object simpliciter: the different C’s should therefore be seen as conditions of being an object and not just as conditions of being an object of experience. Thus, the results that a Kantian really wants to establish are of the following kind:

(3) Necessarily, if x is an object, then x is C.121 The instances of (3), it seems, are like any old necessary truths. That they nevertheless belong to transzendentalphilosophie is due to the fact that they are established by a method that is transcendental. That is to say, “condition of being an object” is explicated through “condition of objective cognition”, i.e., through considering what is involved in object-related judging or judgment. Understood in this way, the establishment of conclusions of the form (3) still involves the characteristically Kantian inference; only it now proceeds from conditions of thought (“objective cognition”) to

121 It seems that Kant himself was less than fully clear about the differences between (1) and (2). Brook (1992) makes the important observation that there is a notorious slide in Kant’s formulation of the central question of the first Critique. In the Introduction he writes as if the question to which he is going to give an answer is: How can there be judgments which are synthetic and nevertheless necessary/apriori knowable? Once he sets out to answer the central question, however, he in fact reformulates it so that it is no longer about synthetic judgments but about us, i.e. about our ability to make judgments with the relevant characteristics; Broad (1978, pp. 5-8) makes a similar point. Clearly, insofar as our focus is on the latter question, it is not prima facie impossible that a legitimate answer to it might be along the lines suggested by (1). However, if our answer does take this form, it looks like a mere confusion to assume that it also answers the second question, or that there is a legitimate transformation of (1) into (2) (or even (3)). However, not everything that Kant has to say about the topic is attributable to this confusion; again, see Brook’s essay for more details. 246 Ch. 3 Russell on Kant conditions of objects of thought. Accordingly, there may be room here for an argument in the style of Russell (and Eberhard): “conditions of thought” – no less than “conditions of experience” – are psychological and hence empirical constraints on cognition, which disqualifies them from acting as a source of genuine necessity. It seems, however, that the current argumentative strategy gives at least some leeway for the Kantian. Here we meet the second line of argument that can be read with some plausibility into Schulze’s defence of Kant. When Schulze argues that there are “predicates” which are necessary for our ability to represent space, his argument is that such predicates have an object-giving function: if we deprive ourselves of these “predicates”, we are no longer capable of thinking geometrical thoughts or making geometrical judgments. Such rules, that is, are established through an analysis of “objective cognition”. For example, we have already seen that, according to Kant, what underlies geometry (i.e., the judgments of geometry) are the simplest operations with the help of which we give to ourselves – construct – the objects of geometrical thought. Such rules are not plausibly construed as high-level empirical generalizations about how human beings think; rather, their more than purely subjective status is grounded in the fact that they are established by an analysis of the content of geometrical judgments. There is at least one respect in which the idea of transcendental grounding is clearly superior to the r-model plus the notion of relative modality as an interpretation (and, possibly, defence) of Kant: it can explain why synthetic necessities are genuinely necessary in the Kant- ian context. Recall van Cleve’s argument against Kant. He claimed to be able to show that Kant’s conception of modality leads to the conclusion that “there is no necessity”. To this the reply was proposed that a distinction ought to be drawn between kinds of necessity. But it was also seen that, if these kinds are elucidated with the help of the notion of relative modality, we must accept the conclusion that their ground (“our nature”, according to the r-model) is, after all, irreme- diably contingent. This conclusion was then seen to be problematic, because Russell’s original objection would apply with equal force to Ch. 3 Russell on Kant 247 the refined position. van Cleve suggests, however, that this untoward consequence could be deflected, if we were in a position to maintain that the cognitive constitution that grounds synthetic necessities (in his discussion, the form of our intuition) is itself necessary. But if this is to be anything more than an ad hoc assertion, some reason ought to be given why this constitution is correctly regarded as necessary in the Kantian context. And one thing seems clear enough: insofar as we interpret real necessity as a merely relative modality, this cannot be done. The culprit here is the ascription of the bi-polar picture of logical modality to Kant. Consider again the case of geometry. According to Kant’s theory of geometry, the conditions of constructibility imply, for instance, that only certain kinds of triangles can be constructed. Here the question arises, what is the force of this “can”? The relative modality reading suggests the answer that it is a limitative “can”. That is to say, the conditions that together define constructibility single out one particular region in the logical space, which itself is “pre-given”. In other words, there is another set rules, of greater generality than the rules of geometrical construction; and it is natural to call these rules “logical”. They fix the most basic sense in which something is to be possible. Logic thus permits a number of mutually exclusive geometrical alternatives (a number of models), of which one and only one accords with the rules of geometrical construction. Now, even if there are good reasons to think that the question “Why is this particular region singled out and not some other?” is somehow inappropriate here, the fact remains that it could have been some other region: there is nothing in the model of relative modality save mere assertion to preclude the possibility that the rules for geometrical construction could have been different.122

122 There is also the further point that if we insist that exactly one of several logically permissible alternatives is really possible, that may look like a brute fact about “us” and our psychological constitution. To take a concrete example, the neo-Kantians of the late 19th century often argued that even though non-Euclidean geometries are possible in some abstract mathemati- 248 Ch. 3 Russell on Kant

If we take seriously the idea of transcendental grounding, a very different picture emerges. On this view, the crucial point about Kant’s notion of real possibility is that the relevant domain of possibilities, like the domain of geometry, is first given through the conditions on constructibility (and these, we should recall, are established through an analysis of “objective cognition” so that the domain of real possibilities coincides with the domain of objective cognition). What according to the previous picture was regarded as a limit within “logical space”, is now regarded as a segment of its boundary line (though what is limited by this boundary should no longer be called logical space, since it is not given or defined by formal logic). Since what is possible simpliciter with respect to the geometrical domain is explained with the help of the notion of geometrical constructibility, the necessities which flow from this notion are genuine or absolute in the sense defined previously; if a proposition, p, is necessary according to the conditions of constructibility, there is no sense of possibility in which p could have been false.123

cal sense, only Euclidean geometry is “really possible”, because it is the only geometry that we can “imagine” (see, e.g., Land 1877). Apart from all other objections that can be brought against this suggestion, there is the obvious point that such a claim, even if it is true, is no more than a “psychological” fact about us, about what we can and cannot do; and this conclusion is only enhanced by the fact that imaginability was usually understood in some fairly concrete, down-to-earth sense. Since this kind of defence of Kant was fairly common in the 19th century, and since we know that the young Russell was familiar with it (see Griffin 1991, Ch. 4.2 for a detailed discussion), it seems not unlikely that Russell’s “psychologistic” or “subjectivist” reading of Kant owes something to it; for a similar point, see Hylton (1990a, p. 77). 123 This account of necessity presupposes that p really is constitutively related to our geometrical thought; that our geometrical thought really does flow – in part at least – from a rule that commits us to accepting p. Subse- quent developments in geometry (and elsewhere) have of course undermined Kant’s specific candidates for synthetic apriori propositions. It remains unclear, however, whether this shows that Kant’s explanatory strategy itself is ill-conceived, or whether the mistake lies only in the specific propos- Ch. 3 Russell on Kant 249

The gist of Kant’s notion of real possibility can be captured by comparing it to the account of logical necessity that Wittgenstein gave in the Tractatus. A recent paper by Peter M. Sullivan on the early Wittgenstein’s solipsism (Sullivan 1996) is useful on this point. Sulli- van discusses Wittgenstein’s idea that the apriority of logic lies in the impossibility of illogical thought. This is first developed in the Note- books and then restated in the Tractatus (5.4731), where it is explained by means of a geometrical analogy:

We cannot think anything unlogical, for to do that we should have to think unlogically. (3.03) To present in language anything which ‘contradicts logic’ is as impossible as in geometry to present by its coordinates a figure which contradicts the laws of space, or to give the coordinates of a point which does not exist. (3.032)

Wittgenstein’s suggestion seems to be that “what can be thought” relates to logic in the same way as “what can be intuited” does to geometry: “that geometry is apriori consists in the fact that nothing contra-geometrical can be intuited” (Sullivan 1996, p. 199). According to Sullivan, however, the analogy breaks down at a crucial point. In geometry, he explains,

we can know (e.g.) that any triangle has angles equal to two right angles because no other kind of triangle can be constructed in intuitive space. The conceptual specification of a concept doesn’t settle this alone. But the requirement to construct this concept in intuition imposes a tighter constraint. In Wittgenstein, the requirement for something to be thinkable is not a tightening up of any broader conceptual space. In thought, the requirement excludes nothing, and in language nothing is excluded but straightforward, ordinary nonsense. (id., pp. 199-200).

As the quotation from Sullivan shows, he understands constructibility in the limitative way. We have already seen that this is unsatisfactory in the Kantian context. However, what he says about Wittgenstein als to synthetic apriori disciplines that he made. I shall not explore this issue here. 250 Ch. 3 Russell on Kant and logic supplies a more adequate picture which can be readily applied to Kant; in the Kantian context the rules of geometrical construction should be given a function that is (almost exactly) the same as the function that Sullivan’s Wittgenstein assigns to logic in the Tractatus. Insofar as this is one’s preferred reading of Kant, one is in a position to argue that Kant did have a real reason to maintain that the rules of geometrical construction are genuinely necessary or that they are necessary in the absolute sense; the requirement that something should be constructible is not a “tightening up of any broader conceptual space”; rules of construction exclude nothing, because they are what make geometrical thought possible in the first place.124

124 Kant’s explanation of the synthetic necessity is “almost exactly” like Wittgenstein’s treatment of logical necessity; this qualification is needed, because Kant’s treatment of thought and its conditions includes the further component of formal logic. If “what can be thought” is explained in terms of constructibility and other comparable conditions, what are we to say about formal logic? Obviously, Kant did not think that formal logic be- longed to the sphere of Wittgensteinian “nonsense”; he did not think that what is beyond the conditions of possible experience is literally unthinkable. In Kant’s philosophy there is therefore room for a sense of “thinking” that is delineated by the rules of formal logic. In this sense we can form a thought of a thing, x, which does not comply with this or that (or indeed any) condition of possible experience. But the important point is that the notion of possibility that features in such a thought is purely negative: we can think of x, but only in the sense that we cannot derive a contradiction from the concept of x. Since all object-related thought must be mediated by sensibility, this sense is, indeed, very minimal (cf. Gardner 1999, p. 280); in particular, this sense must be kept strictly separate from the idea that we are likely to attach to it. For us the expression “no contradiction can be derived from the concept of x” has a clear positive sense; it suggests the availability of a consistent theory of x and hence the existence of a model. But this positive sense of “thinkable without contradiction” was not Kant’s; as we have noted on several occasions, its development presupposes conceptual tools that were not available to him and were created, for the most part, only after his death. (This does not imply that the idea of a thing that does not conform to the conditions of experience could not have a positive function elsewhere in Ch. 3 Russell on Kant 251

I conclude that Kant’s theory of real modalities can be defended aga inst a number of initial objections, among them Russell’s charge that the proposed explanation of necessity is in principle mistaken, since it implies a confusion of the objective (the necessities that the theory seeks to explain) with the subjective (the proposed explanatory basis). It is a familiar view that an objection to Kant like the one that Russell advances is based on a psychological or subjectivist interpre-

Kant’s theoretical or practical philosophy. Obviously it can and does; this, however, does not undermine the present point.) This is in my opinion the most plausible reading of Kant’s well-known discussion of the concept of a biangle (A220/B268). Kant says that he concept of a figure which is enclosed within two straight lines contains no contradiction, because the concepts of two straight lines and of two straight lines’ coming together “contain no negation of figure”. This concept, however, is ruled out by the “conditions of space and its determinations” (ibid.) This passage can be read in two ways. Firstly, it may be argued that the passage shows that Kant appreciated the fact that non-Euclidean geometries are consistent; this is how Martin (1955, pp. 23-25) and Brittan (1978, p. 70fn. 4) read it. The second possibility is to see Kant’s remark in the light of the above discussion: the point Kant is making in the passage would then be only that the concept of a biangle is possible in the negative and minimal sense. That the second reading is correct has been recently argued for by Judson Webb (1995). His reasons, though, are rather different from those given above. Very briefly, his argument is that Kant could have had evidence for the consistency of a non-Euclidean geometry only if he possessed at least a rudimentary notion of a model, but that “there is not the faintest suggestion of any such notion” anywhere in Kant’s writings (id. p. 2). Webb’s argument for this conclusions is based on a very sophisticated discussion of Lambert’s 1786-memoir on parallel lines, Lambert’s work being the only possible source from which Kant might have acquired information about non-Euclidean geometries. Needless to say, what has been said above is perfectly consistent with Webb’s more technical discussion. Moreover, the interpretation of Kant’s philosophy of mathematics that was presented in chapter two lends support to Webb’s argument; the important thing is of course the relatively straightforward observation that Kant’s notion of formal logic was far too weak to support even the most rudimentary notion of model. 252 Ch. 3 Russell on Kant tation of transcendental idealism. This is of course correct to the extent that the r-model regards synthetic apriori judgments as consequences of certain very general facts about the constitution of our minds, and such facts, we may admit, can be described as “psychological”. We have seen, however, that the Argument from Necessity has another – and, in my opinion, more interesting – presupposition lurking behind it. This is the notion of relative modality, which is often assumed as the more or less implicit starting-point when Kant’s views on modalities are discussed. Russell himself is committed to it, when he interprets Kant’s explanation of the synthetic apriori in terms of the r-model; but Kant-scholars, too, accept it to the extent that they attribute the bi-polar picture of logical possibility and necessity to Kant. The Argument from Necessity purports to question the explanatory adequacy of the Kantian theory of necessity on certain very general grounds which have to do with the explanatory strategy underlying the r-model. As I have presented it so far, the argument trades on the alleged contingency of our mental constitution and the consequences that this has for its capability to figure in explanations of necessity. Occasionally, however, Russell has in mind slightly different considerations, which can be considered preliminary to the issues concerning contingency. Putting these together yields a two-stage argument against the Kantian theory. The general form of the two-stage argument has been discussed in some detail by Simon Blackburn (1986, pp. 120-121). He regards it as a serious problem for any broadly truth-conditional account of necessity. What we wish to explain is necessity (usually of this or that specific kind, so that the focus is on logical or moral or mathematical or natural necessity, etc. through the whole range of kinds of necessities that could be distinguished). Our curiosity as to why q is necessary may be satisfied with an explanation that takes the form of a “local proof” of q from p (id. p. 120). But this will not do as a general model for such explanations. There are two reasons for this. Firstly, the local proof simply assumes that p is the case and invests it with no necessity; hence the proof is satisfactory qua explanation only to the extent that Ch. 3 Russell on Kant 253 we antecedently understand why p is necessary. In philosophy, however, we are concerned not so much with particular instances as with kinds of necessity, so that what we are looking for is a generic explanation. Hence, we must shift attention to our understanding of the putative source of the necessity in question. Secondly, the inferential principles that underwrite the local proof of q from p are themselves simply assumed, so that the acceptability of the explanation is conditional on our understanding of the status of (logical) proof. Russell applies these points to the Kantian theory in sections 430 and 431 of the Principles. He argues that Kant’s theory of necessity, since it complies with the r-model, cannot deliver a general (or exhaustive) explanation of necessity. The Kantian wishes to show that a proposition about the nature of space is necessary by deriving it from a proposition describing the constitution of our minds. Adopting this strategy, however, “we only push one stage farther back the region of ‘mere fact’” (§430). That is to say, propositions describing the constitution of our minds are mere premisses for the proof: these premises, together with the rules of inference, are merely assumed, as Russell explains in the next paragraph. We are thus lead to ask what makes for the necessity of the premises (we may bracket questions about the status of proof here). This question moves us on to the second stage of the argument. Again, Blackburn formulates the problem at a general level. It takes the form of a dilemma. One’s preferred explanation – very roughly, one’s preferred way of spelling out the truth-conditions of statements featuring modal-operators – is either in terms of what is the case, or else in terms of what must be the case. In the former case, one is likely to meet the objection that the explanation undermines the necessity of explanandum; what merely is the case could have been otherwise. Hence the putative explanation is no explanation at all. In the latter case, one is likely to encounter the charge of circularity: explaining necessity in terms of what must be the case is unsatisfactory, because “there will be the same bad residual ‘must’” (Blackburn 1986, p. 120). At the very least, this horn of the dilemma suggests that an explanation that 254 Ch. 3 Russell on Kant remains within the modal sphere cannot be accepted as ultimate, but must be considered incomplete. Blackburn illustrates the first horn of the dilemma by a familiar example. A modal theorist might hold that twice two is four, because our linguistic conventions make it so (some such theory was popular in the early decades of the 20th century). If, furthermore, an advocate of this theory refuses to attach any kind or degree of necessity to our conventions, there arises, in Blackburn’s words, a “principled difficulty about seeing how the kind of fact cited could institute or be responsible for the necessity” (id. p.121): if that really is all there is to necessity, then twice two does not have to be four. The substitution of our mind and its constitution for linguistic conventions brings us to Russell’s second point; if the facts cited are “mere facts” about the constitution of our minds, reference to them seems to undermine, rather than explain, the original modal status of the explanandum; if the constitution of our mind is all there is to the necessity of twice two is four, then twice two does not have to be four. (We have already seen that this argument has a non-trivial presupposition; the necessity to be explained must be one that is plausibly construed as being of the absolute kind, so that its relativisation to something that is modally unqualified has the prima facie effect of undermining it. If, on the other hand, the explanandum is of a sufficiently modest type, no difficulty arises in this respect.) The dilemma that Blackburn pinpoints is perfectly general, as it can be raised against any theory of modality that operates with what could be called the “generalized r-model”: ‘ɷp, because F’. Here F can be anything (modal or non-modal) that can be cited, roughly, as the determiner of modal truth (the constitution of our mind, linguistic conventions, God’s will, a plurality of maximal mereological sums of spatiotemporally related individuals, etc.), and “because” is to be understood in a way that is sufficiently flexible so as to allow a number of distinct truth-driven projects to be subsumed under it. Summing up our discussion so far, the Argument from Necessity is to be understood as the following two-stage argument. The first stage consist in the observation that the derivation of necessity from Ch. 3 Russell on Kant 255 the constitution of our minds cannot constitute a complete general explanation of necessity: a proof of necessity that is merely local leaves unexplained the necessity of this constitution and of the principles underwriting the derivation. The second stage focuses on the modal status of the constitution: since it is contingent that our mental make-up is what it is, it cannot be the source of genuine (or absolute) necessities. As far as I know, Russell nowhere discusses Blackburn’s dilemma in its general form. In particular, he does not discuss the second horn of the dilemma, the case where the explanans, F, is construed as necessary, rather than contingent. He ignores the second horn, because he thinks that Kant’s F – the constitution of our minds – is a “mere fact”, i.e., that it is genuinely contingent; it is therefore only the first half of the dilemma that matters for his purposes. It would, of course, be possible for a defender of Kant to question this assumption of Russell’s, arguing that when what is at stake is the constitution of our minds in the sense that is relevant for the explanation of the synthetic apriori, that constitution must be taken to be necessary, rather than contingent. It might then be suggested that the proper model to be invoked in an attempt to understand Kant’s explanation is the modal variant of the generalized r-model. There are obvious difficulties with this suggestion, however. Firstly, it seems to be entirely ad hoc. Secondly, even if reasons could be given for the view that the constitution is a matter of necessity, it seems that reference to that constitution, insofar as it is understood in accordance with the generalized r-model, can do no more than explain necessity in some psychological sense. Thirdly, there is the question as to how we are supposed to attain knowledge of the putative explanatory basis; obviously, this question arises whether the basis is necessary or merely contingent.125 Such difficulties, it seems, can be circumvented only by a sufficiently radical departure from the r-model. We have already seen that Kant’s notion of real possibility, once it is properly expounded,

125 Russell raises this question at (1911a, p. 39). 256 Ch. 3 Russell on Kant points in this direction. A present-day Kant-scholar is, indeed, likely to argue that the Argument from Necessity cuts no ice with Kant, because any interpretation of transcendental idealism that uses the r- model can only be described as a travesty of his true intentions; in particular, a defender of Kant will probably issue the complaint that the argument assumes a much too “subjectivist” or “psychological” reading of transcendental idealism. This reading, it may be admitted, receives some superficial support from Kant’s frequent use of psychological or quasi-psychological phrases, claims and distinctions; it is argued, nevertheless, that what Kant was really doing was not psychology in the sense intended by his critics, but something – be it epistemology or transcendental psychology or something else – that is immune to the kinds of charge that Russell, among others, has directed against it.126

126 Allison (1983, pp. 6-13) is a good example of this strategy. He dismisses the standard picture (and therefore, also, Russell’s interpretation) on the grounds that it fails to observe a number of crucial Kantian distinctions; in particular it ignores the decisive difference between empirical and transcendental versions of the distinctions between ideality and reality, on one hand, and between appearances and things in themselves, on the other. In Alli- son’s own terminology, the standard picture fails to grasp Kant’s most fundamental innovation, the notion of an epistemic condition. This expresses the idea that there are conditions “necessary for the representation of an object or an objective state of affairs” (id., p. 10). The term “epistemic condition” is intended to cover both the pure concepts of the understanding and the forms of human sensibility (space and time). Armed with this notion, Allison goes on to suggest that this genuinely Kantian notion is “frequently confused” with other senses of “condition” (id., p. 11). In particular, it is confused with the notion of a psychological condition, by which Allison means “some mechanism or aspect of the human cognitive apparatus that is appealed to in order to provide a genetic account of a belief or an empirical explanation of why we perceive things in a certain way” (ibid.) Clearly, this criticism, if well-founded, applies to Russell’s use of the r-model in his interpretation of transcendental idealism. It may be observed that the sorts of considerations that Allison presents could be used to dismiss Moore’s and Russell’s direct arguments against transcendental idealism. Let us consider just one example. It may be argued, Ch. 3 Russell on Kant 257

Naturally, such claims cannot be evaluated here, as that would lead us too far into the details of Kant’s philosophy. For our purposes the important point is that the r-model is in fact everywhere presupposed in Russell’s discussion of Kant; it is the linchpin of Rus- sell’s interpretation and criticisms of Kant’s explanation of the synthetic apriori. Our discussion of the Argument of Necessity has shown, however, that Russell does succeed in making points that are largely independent of the issue of psychologism. The r-model does have (at least) one genuine interpretative virtue: it enables us to raise the question concerning the model for modalities that underlies Kant’s notion of real possibility and real necessity. I shall next consider Russell’s second and third objection to Kant’s explanation of the synthetic apriori, the Argument from Truth and the Argument from Universality. Again, these flow fairly directly as against Russell, that Kant’s theory is not to be seen as an error-theory: when we endorse an ordinary proposition, x, about the properties of space, we speak what Allison calls the “language of experience” (id., pp. 8-9). This language “includes both ordinary and scientific experience”. Construed in this way, x is about and is made true (or false) by the character of the space, and as long as we remain at the empirical level (or speak the language of experience) it is simply false to say that x has the truth-value that it does have in virtue of the mind. But there is another level of description, one that must be kept strictly separate from the empirical level. This is the level of transcendental reflection. It is only when we ascend to this level (or adopt this new standpoint) that assertions about space “become” dependent upon the character of the mind. Hence, there is a sense in which x is about space and is made true or false by the properties of space; this sense is the empirical sense. But there is also another sense in which x is about the mind; this is the transcendental sense. Therefore the two propositions “x is about the space” and “x is about the mind” are perfectly compatible. Whether this rejoinder to Russell is adequate is a question which must be regarded as open, I think. The answer to it depends upon whether coherent sense can be made of the distinction that Allison argues is in the heart of transcendental idealism. I shall not try to decide the issue here, although I am inclined to think that Allison’s distinction between different levels or standpoints or languages or manners of considering (about which more much ought to be said) is obscure at some quite fundamental level. 258 Ch. 3 Russell on Kant from the r-model. Since the model has already been discussed in considerable detail, the other two objections can be dealt with relatively briefly.

3.6.3.3.3 Another Argument from Necessity

In discussing the Argument from Necessity, I ignored certain complications regarding the r-model. I argued that Russell’s argument is best seen as a special case of a perfectly general worry about explanations of necessity of a certain particular type, to wit, such as belong to a family of truth-conditional accounts. Consequently, I assumed that the r-model (hence, Russell) attributes to Kant a truth-conditional account of the source of synthetic necessities. This, however, is not the only way that the r-model can be interpreted. And it is not the only way Russell always reads Kant; in fact, it is probably not his standard reading. In section 3.6.3.2 I formulated two anti-Kantian arguments both of which can be read with some plausibility into §430 of the Principles. According to the first argument, the transcendental idealist theory of space entails that we are systematically mistaken about the truth- makers of propositions about the nature of space. The second argument had its focus on the characteristically Kantian notion that synthetic apriori propositions are true in virtue of or made true by the constitution of our minds, the claim being that these phrases, as they are used in the Kantian context, can only mean that our mind is so constituted that it makes us think in a certain way. The r-model can be so understood that it captures just this latter idea. It would then give rise to an Argument from Necessity that is rather different from the one discussed above. Such an argument would start with the assumption that a proposition’s necessity is simply a matter of our being so constituted that we must think in a certain way. From this premise one could then derive a number of untoward consequences. At least some of the points that Russell makes by way of criticizing Kantian theories of necessity can be understood along Ch. 3 Russell on Kant 259 these lines. It is likely that he did not distinguish very sharply between these two interpretations. It is for this reason that I have concentrated on the truth-conditional version of the Argument from Neces- sity, which, furthermore, is philosophically by far the more pregnant.

3.6.3.3.4 The Argument from Truth and the Argument from Universality

The situation is more straightforward with the other two indirect arguments, the Argument from Truth and the Argument from Univer- sality. Russell’s use of these arguments is predicated on the non-truth- conditional version of the r-model. To repeat, the claim is that synthetic apriori judgments owe their special status to our psychological constitution; they are, to use Broad’s formulation, necessary consequences of certain facts about the human mind. This assumption is part and parcel of the standard picture of transcendental idealism. It is usually formulated with the help of the notorious coloured spectacles -simile, but the connection between this formulation and the one in terms of propositions and truth is immediate as is made evident by the following passage by Russell:

Idealism – at least every theory of knowledge which is derived from Kant – assumes that the universality of à priori truths comes from their property of expressing properties of the mind: things appear to be thus because the nature of the appearance depends on the subject in the same way that, if we have blue spectacles, everything appears to be blue. The categories of Kant are the coloured spectacles of the mind; truths à priori are the false appearances produced by those spectacles (Russell 1911a, p. 39).)

Understood in this way, synthetic apriori propositions appear to have no other warrant than their universal existence (as Russell wrote in his review of Poincaré), or as we might put it, their forced acceptance: they are propositions which we cannot but believe, given our mental constitution. 260 Ch. 3 Russell on Kant

The import of the Argument from Truth is as simple as it is obvious: Kant’s explanation of the synthetic apriori can offer no genuine warrant for the relevant propositions. There simply is no internal connection between accepting and rejecting a proposition, on the one hand, and its truth and falsehood, on the other. In particular, a proposition which is universally accepted – even if this acceptance is forced upon us – may nevertheless be false. Russell makes this point in the Problems of Philosophy, using the law of contradiction as his example. He points out that what we believe when we believe the law is “not that the mind is so made that it must believe the law of contradiction” (1912a, p. 88). The law, he emphasizes, is about things and not only about thoughts: “[i]f this, which we believe when we believe the law of contradiction, were not true of things in the world, the fact that we were compelled to think it true would not save the law of contradiction from being false; and this shows that the law is not a law of thought” (id. p. 89). Normally one would think that if a proposition’s only warrant is its universal, or forced, acceptance, no conclusion can be drawn as to its truth-value: for all we know, it may be true, but it may be false as well (even if this latter option should be unthinkable). This, however, is not how Russell states the conclusion of the argument. For he says categorically that we know that synthetic apriori propositions, when they are construed in the Kantian manner, are false; that they are nothing but “universal delusions”, produced by the coloured spectacles of Kantian categories and forms of intuition. Russell is entitled to this stronger conclusion, given his subscrip- tion to the standard picture of transcendental idealism. This implies, among other things, the view that Kant’s appearances relate to things-in-themselves as representations do to their objects. When this premise is paired with the further assumption the things-in- themselves are unknowable (as Kant thought), it follows – given a few auxiliary assumptions about knowledge – that we do not really know anything at all. Russell is thus in agreement with Moore’s and Prichard’s construal of the distinction between appearances and Ch. 3 Russell on Kant 261 things-in-themselves;127 if appearance-talk means that we only know of “things” how they seem to us, and not how they really are, the proper conclusion to be drawn from this is that we do not really have any knowledge at all of, because to know something is to know it as it really is. This construal of the appearance vs. thing-in-itself distinction is also found underlying the Argument from Universality. Kant had argued that strict or true (as opposed to “assumed and comparative”) universality is one of the marks of apriori propositions; such a proposition is one which is thought “in such manner that no exception is possible” (B4). Since appearances and things-in-themselves constitute two disjoint classes of objects, the restriction of our so-called “knowledge” to appearances has the consequence that synthetic apriori propositions are not really universal at all, contrary to Kant’s explicit claims; things-in-themselves are outside their scope. Russell argues, as against Kant, that reflection on arithmetical propositions reveals no such limitation in their scope and no difference in their applicability to appearances and things in themselves (“physical objects” according to Russell’s reading of Kant):

[i]f there is any truth in our arithmetical beliefs, they must apply to things equally whether we think of them or not. Two physical objects and two other physical objects must make four physical objects, even if physical objects cannot be experienced. To assert this is certainly within the scope of what we mean when we assert that two and two are four. Its truth is just as indubitable as the truth of the assertion that two phenomena and two other phenomena make four phenomena. (1912a, pp. 87-88)

We may sum up Russell’s three indirect anti-Kantian arguments in the following manner. Starting with the assumption that transcendental idealism can be explicated with the help of what has been referred to as the r-model, Russell finds Kant’s explanation of how there can be synthetic apriori propositions multiply flawed. According to the r- model, synthetic apriori propositions have their foundation in facts

127 Cf. section 3.6.2.2. 262 Ch. 3 Russell on Kant about the constitution of our minds. These facts, though general, are nevertheless ultimately on a par with any other facts about the natural order. Hence, they cannot provide a foundation for the synthetic apriori (like the propositions of mathematics). Firstly, they are contingent, and cannot therefore account for the necessity that we attach to mathematics (and even if they were necessary, this necessity, being no more than psychological, would not be of the right kind). Secondly, citing such facts cannot really explain how the relevant propositions can be true. What the r-model can accomplish is at most a description of the causal-cum-psychological origin of certain beliefs, and hence it replaces truth properly so-called by the surrogate notion of universal acceptance. Thirdly, reference to mental facts is unable to account for universality in the required sense (“strict universality”), substituting something considerably weaker for it. The early Russell completely rejected the r-model as an explanation of the synthetic apriori. But he did not suggest any alternative which could take its place. In the Principles, §10, we find the following comment:

The fact that all mathematical constants are logical constants, and that all the premisses of mathematics are concerned with these, gives, I believe, the precise statement of what philosophers have meant in asserting that mathematics is à priori.

It would be natural to assume that this statement forms the first step towards a new theory of the apriori in mathematics. This, however, is not the case, insofar as “apriori” is understood in the usual way, viz. as a label for an epistemic category. For Russell shows, at this stage of his career, very little interest in epistemology. The motive underlying Kant’s theory of the synthetic apriori was to find an explanation of how there can be non-trivial (that is, synthetic) extensions of knowledge which are nevertheless apriori and hence not dependent by way of their justification upon sensory input. Kant argued that this problem can be resolved only on the assumption that, at some fundamental level, the objects of knowledge cannot be regarded as fully inde- Ch. 3 Russell on Kant 263 pendent of the knowing subject. As we have seen, Russell rejected this conclusion for a number of reasons. But he did not offer as much as a sketch of an alternative account which would have included an assumption of independence such as would have contra- vened Kant’s Copernican turn. Such considerations play no role in the overall argument of the Principles; this argument, we have seen, is more concerned with content or meaning, i.e., with the objects of mathematical understanding. There is, then, a clear sense in which Russell’s criticisms of Kant are incomplete; he does not address the epistemological questions which were Kant’s focus and which more recent philosophers have made one of the central issues in the philosophy of mathematics.128 Russell thus ignores the question: How is pure logic possible?129 It is obvious, however, that this does not prevent him from having all sorts of more or less articulated ideas about “pure logic”, including its epistemology. It is to these, i.e., Russell’s conception of logic, that I shall turn next.

128 What I have in mind is the well-known difficulty of accounting for our epistemic access to the entities that figure in the explanation of mathematical truth; the classical discussion is Benacerraf (1973). 129 Cf. de Laguna (1907). See however, the end of chapter 5 for a somewhat different perspective on this issue.

Chapter 4 Logic as the Universal Science I: the van Heijenoort Interpretation and Russell’s Conception of Logic

4.0 Introduction

In 1967, Jean van Heijenoort published a short paper, “Logic as Cal- culus and Logic as Language”. In the paper he argued that we gain important insight into the early history of mathematical logic – the period that extends from Frege’s Begriffsschrift of 1879 to the early 1930s – if we distinguish between two radically different ways of looking at logic. According to this interpretative framework, the conception of logic as calculus relies on model-theoretic conceptualisations, whereas the view of logic as language (or the universalist conception of logic) sees logic more as a framework for all rational discourse, or at any rate, as a skeleton for such discourse. Central to the van Heijenoort interpretation – and to the criticisms that have inevitably followed – is the view that the conception of logic as language excludes a metatheoretic perspective on logic. Since model-theoretic conceptualisations are essentially metatheoretic, the question whether or not such perspective is available has far-reaching consequences for logical practice and for any issue to which this practice is relevant: an example would be the question concerning the proper formulation of mathematical theories. Russell is commonly regarded as an advocate of the universalist conception. Although this view is undeniably correct in some sense, it is far from a straightforward task to determine what this conception implies and what it excludes. In this chapter I will examine the issues pertaining to the van Heijenoort interpretation. These revolve around the question “metatheory vs. no metatheory”, and its application to the early Russell. The early Russell’s understanding of logic is grounded in a conception of proposition that is intended to provide a general framework for logical analysis comparable not only in its function but also in 266 Ch. 4 Logic as the Universal Science I many of the more important details to Frege’s functional analysis of content. We shall also see that his version of the universalist conception can be traced to this notion of proposition as a non-linguistic – more generally, non-representational – entity. This view has a couple of important consequences. It implies, firstly, that semantic questions cannot have that importance to the early Russell that they have for an advocate of the model-theoretic conception. This point can be illustrated by how he construes the notion of truth; his “anti- semanticism” does not mean that there is no room for a concept of truth, but it does mean that the role of the concept is metaphysical, rather than semantic. Secondly, the view also implies that when Rus- sell is thinking about logic, what he has in mind are the principles underlying correct reasoning. It might be thought that it is precisely this view of logic that excludes metaperspective. Closer inspection of the relevant issues suggests, however, that this conclusion cannot be upheld. There are good reasons to think, then, that the question whether or not metaperspective and metatheoretic investigations are possible is not really the key to the universalist conception. In addition to this conclusion, we shall also see that Russell’s actual logical practice contains elements that are not at all easily reconciled with his official metaphysical commitments. In particular, when we examine how he understood pure mathematics, we shall find that the true picture is a good deal more complicated than is suggested by any straightforward application of the van Heijenoort interpretation.

4.1 “Logic as Calculus and Logic as Language”

“Logic as Calculus and Logic as Language” has become a source for an entire tradition in the historiography of modern logic.1 According

1 Another source, apparently, is Burton Dreben’s activity at Harvard from the 70s onward; see Tappenden (1997, p. 220). According to Peckhaus (2003, p. 5), we should distinguish between historians’ and non-historians’ “common understanding of the early directions in modern logic”. According Ch. 4 Logic as the Universal Science I 267 to van Heijenoort, we can better appreciate how Frege saw his own work by considering what is involved in his claim that the aim of his concept-script (Begriffsschrift) was to develop not merely a calculus ratiocinator but also (and primarily) a lingua characterica.2 van Heijenoort argues, furthermore, that the contrast to which this distinction gives rise – the conception of logic as calculus and the conception of logic as (a universal) language – can be used to elucidate the early development of mathematical logic in its entirety; the time-span which he discusses stretches from Frege to Jacques Herbrand’s thesis in 1929. Central to the idea of lingua characterica is the feature that van Hei- jenoort calls the universality of logic. This in fact amounts to several things. First of all, it has to do with the differences between quantification theory and propositional calculus; the latter represents atomic sentences as mere truth-values, whereas the former can articulate their content. In this way, logic is transformed into a genuine language – lingua – by dint of which entire stretches of scientific knowledge could perhaps be formulated. For Frege, the important field of application is of course mathematics (arithmetic). There is more to the universality of logic than this, however. In Frege’s concept-script universality has to do with a central feature of the system: quantifiers binding individual variables range over all objects (and, we should add, functional variables range over all functions modulo the stratification of functions into different levels). Unlike the likes of De Mor- gan and Boole, for whom, according to van Heijenoort, the domain to Peckhaus, van Heijenoort is largely responsible for today’s received view of the development of modern logic, but he adds the warning that van Hei- jenoort’s can be called the received view only among non-specialists (id., p. 4). Since I am not concerned with history in the way that Peckhaus, probably, is, I shall largely ignore this complication. What is clear – and Peckhaus readily admits this much – is that van Heijenoort’s views and their further refinements have been very influential in recent discussions of the philosophies behind the development of modern logic. 2 See Frege (1882). Frege made his remark by way of a polemic against Ernst Schröder’s review of the Begriffsschrift; see the introduction to Frege (1972). 268 Ch. 4 Logic as the Universal Science I of quantification or universe of discourse could be changed at will, Frege has only one domain of quantification: “Boole has his universe class, and De Morgan his universe of discourse, denoted by ‘1’, but these have hardly any ontological import. They can be changed at will. The universe of discourse comprehends only what we agree to consider at a certain time, in a certain context. For Frege it cannot be a question of changing universes” (van Heijenoort, p. 325). More precisely, his procedure is not a restriction to one universe, but a conviction that in the last instance quantification takes place with respect to the universe; better still, he construes quantifiers as ranging over everything there is (again, this statement must be qualified in view of the distinction between objects and functions and the stratification of functions into different levels, but this does not affect the basic point). As van Heijenoort points out, what this everything includes is a further and independent issue; for example, Frege believed that there are entities besides the ordinary physical objects. The universalist conception has several important consequences for logic. Three such consequences are mentioned by van Heijenoort. Firstly, it follows from Frege’s views on quantification that functions must be defined for all objects (id., p. 326). And when Frege deals with some special domain of objects, he uses methods that are equivalent to the relativization of quantifiers. Secondly, the universality of logic implies that “nothing can, or has to be, said outside the system” (ibid.) More precisely, metalogical questions which require for their formulation the possibility of reinterpretation (questions like completeness, consistency and independence of axioms) are never raised by Frege. Frege, van Heijenoort admits, is not unaware of the fact that a formal system must have rules that do not belong within the system. Such rules, however, are simply “rules for the use of our signs”. These signs do not have any intuitive logic; the only thing that can be done with them is their manipulation according to fixed, purely syntactic rules. And this is precisely the reason why formalisms are useful, according to Frege (ibid.) This is not the only sense of “universality” that is relevant for the Ch. 4 Logic as the Universal Science I 269 view of logic as language. van Heijenoort does not offer an explicit formulation of the other sense, but he does mention one of its consequences, when he makes a comment that is somewhat mysterious at first reading: “[s]ince logic is a language, that language has to be learned. Like many languages in many circumstances, that language has to be learned by suggestions and clues” (ibid.) No explanation is given why logic has to be learned in this way, and there is nothing in the universality as he has explained it that would necessitate such a view. But the thesis about learnability does follow, if we assume that logic is universal in another sense than that specified so far; the key idea here is that logic is universal in the sense that there is but one logic. No matter how many different languages there may be, they all share one and the same logic (“the only language that I understand”, as Wittgenstein put it in the Tractatus3). One may, of course, try to make this logic explicit; this is precisely what Frege, Russell and the other universalists tried to do. And insofar as one succeeds, the result is to be regarded as a streamlined version of a natural language fragment, that is, a system in which concepts and propositional contents can be represented in a perspicuous manner. Now, this supposedly universal logic cannot really be conveyed to someone through a use of language, because the mediating language can only be understood if one already understands its logic, which ex hypothesi, is identical with the one and only universal logic. This “conveyed”, which is excluded by the universalist conception, must, of course, be taken in some sufficiently strong sense. For there are weaker senses which are available for the universalist. Frege sometimes points out that the statements which he makes when he introduces and discusses his concept-script are inaccurate; that, if understood literally, they miss his point.4 He therefore asks the reader not to deny him a share of good will, to come to meet him halfway.

3 Wittgenstein (1922) 5.62. 4 See, for example, “On Concept and Object”, where Frege explains that what is logically simple cannot be defined, but can only be introduced with the help of what he calls an “elucidation” (Frege 1892, p. 182-183). 270 Ch. 4 Logic as the Universal Science I

In the end, one can get inside Frege’s language only by learning how to use it, and it is only through the use of that language that certain otherwise elusive “metasystematic” features – like an explicit formulation of the distinction between objects and functions – show themselves for what they really are. Having stated these consequences, van Heijenoort goes on to argue that Frege’s presuppositions were shared by the authors of Prin- cipia Mathematica. van Heijenoort argues, firstly, that the introduction of logical types necessitates a modification in the original universality thesis. Secondly, he explains why Whitehead and Russell failed even to raise the question that, as Gödel put it, “at once arises”, when one sets up a system like that of Principia, namely, whether the proposed system of axioms and rules of inference is complete.5 According to van Heijenoort, the reason why Russell and Whitehead failed to ask this question is that they implicitly accepted the essentials of Frege’s conception of logic. Thus, metasystematic questions, including completeness in Gödel’s sense, are outside the purview of logic. The only question of completeness is an experimental one, i.e., whether the proposed system of axioms and rules can exhaust the intuitive modes of reasoning that are actually used in the field the system is intended to capture. van Heijenoort contrasts the Frege-Russell(-Whitehead) approach with Leopold Löwenheim’s 1915-paper on the calculus of relations. In distinction to the Frege-Russell approach, which could be called axiomatic, van Heijenoort suggests that Löwenheim’s approach might be dubbed set-theoretic. Löwenheim has neither axioms nor rules of inference, but formulates his logic using naive set theory. Provability is no longer the key question; it is replaced by the questions of whether certain specified kind of well-formed formulas are valid in different domains. Löwenheim was able to establish several important results about his calculus of relations, results that belong to what

5 Gödel’s remark is from his 1930-paper on the completeness of quantification theory; see Gödel (1930, p. 583). Ch. 4 Logic as the Universal Science I 271 we know as first-order model-theory.6 Such results were wholly alien to Frege’s and Russell’s approach to logic. Initiating systematic, serious study of model theory, Löwen- heim’s Über Möglichkeiten im Relativkalkül implied not only a sharp break with the Frege-Russell approach; it also forged a connection with pre-Fregean modes of logical thought. Löwenheim not only used Schröder’s notation for the algebra of logic; he reintroduced the idea of changing the universe at will and basing important considerations on this idea. The interpretative framework suggested by van Heijenoort has inspired a considerable amount of further research, some of it inevitably highly critical.7 Where van Heijenoort gave at best only a sketch of the two positions and their respective consequences, later scholars have worked them out in considerable detail. The bulk of this research has concentrated on attempts to understand Frege and his conception of logic, and other alleged proponents of the view of logic as language have received less attention. Although Russell is habitu- ally classified as an advocate of the conception, his case has been analysed in detail only by Peter Hylton.

4.2. van Heijenoort’s Distinction

4.2.1 The Technical Core of the Model-theoretic Conception

I shall now turn to a more detailed discussion of van Heijenoort’s

6 See Badesa (2004) for a detailed study of Löwenheim’s work in model theory. 7 The relevant literature includes at least the following items: Goldfarb (1979), (1982), (2001), Hylton (1981), (1991a), (1991b), Ricketts, (1985), (1986), (1996), (1997), Kusch (1989), Conant (1991),Weiner (1990), (2001), Kemp (1995), (1996), (1998), Stanley (1996), Tappenden (1997), Landini (1998, Ch. 1), Floyd (1998), Alnes (1998), Goldfarb (2001), Shieh (2001) Peckhaus (2003). Hintikka’s numerous papers on the topic have been collected into Hintikka (1997); for a representative sample, see Hintikka (1988). 272 Ch. 4 Logic as the Universal Science I distinction and its numerous implications. In what follows, I shall speak of “universalist” and “model-theoretic” conceptions of logic (cf. Hintikka 1997, p. xi). The latter term may be surprising at first hearing; after all, model-theory in its present-day form dates only from the 1950s.8 However, recognizably model-theoretic techniques were developed by mathematicians and logicians already in the 19th century,9 and the features which, according to the scholars in the van Heijenoort tradition, most pregnantly distinguish the calculus conception from the universalist one are precisely those that have to do with the legitimacy or otherwise of model-theoretic techniques and conceptualisations. It is advisable to begin the discussion with a comparison of the two positions that is, from the scholarly point of view, as neutral as possible. For it is an undisputable fact about the early Russell that his way of looking at the formulas of his logic is non-trivially different from the currently standard model-theoretic conception.10 Both the universalist and the model-theoretic conception possess what could be called a technical core. Insofar as we confine ourselves to explicating this core, we remain on relatively neutral ground. Scholarly dispute begins only when one moves on from this ground to consider the putative wider implications of the two conceptions. In particular, we are led to consider the implications that the technical core of the universalist conception has for such key concepts and

8 For the development of model-theory, see Vaught (1974) and Badesa (2004). 9 And perhaps even earlier; see Webb (1995). 10 Here and elsewhere in this chapter I make the following two assumptions. I shall assume, firstly, that there is a present-day conception of what logic is and what it does that can be called a “model-theoretic” conception. Secondly, I shall assume that there is a sufficient consensus on the essential content of the model-theoretic conception among those who practice model theory and those otherwise sufficiently qualified to issue a judgment on such questions for that conception to be one, rather than many. Besides the model-theoretic one, there are probably other “conceptions” of logic currently available. They are of no concern here, however.) Ch. 4 Logic as the Universal Science I 273 ideas as, for example, truth, semantics, metatheory and necessity; these will be addressed in section 4.3. The technical core of the model-theoretic conception may be called the schematic conception of logic (I borrow this term from Goldfarb 2001, p. 27). It can be summarized by the following nine points; I follow, by and large, Goldfarb’s (id., pp. 26-7) exposition, with a few modifications:11 [1] The subject-matter of logic consists of the logical properties of sentences and the logical relations holding between sentences. [2] Sentences have logical properties and stand in logical relations to other sentences in virtue of possessing logical forms. [3] Logical forms are representations of the composition of sentences. They are constructed from logical signs (“logical constants”) using schematic letters of various kinds. The description of how this composition is effected belongs to logical syntax (or simply syntax), and when it is done systematically, it constitutes a recursive definition of the notion of a well-formed formula. [4] Syntax may also provide a proof-procedure, which consists of rules of inference plus, possibly, axioms. [5] The well-formed formulas specified in syntax are schemata. They do not state anything and are therefore neither true nor false. They can, however, be interpreted. When a schema is interpreted, a domain of objects (“universe of discourse”) is assigned to the quantifiers, predicate letters are assigned extensions over the domain, individual constants are assigned members of the domain, etc. When a schema is interpreted, it receives a truth-value under an interpretation. This is the central concern of logical semantics (or simply semantics). [6] Semantics thus delivers a truth definition for the formal language under study. It is a recursive specification of the notion of “true in a model” (or “true under an interpretation”: the terminology is not uniform). A model is a set of objects (“the universe of discourse”), together with an interpretation function, which maps the non-logical

11 Goldfarb’s presentation is based on Quine (Quine 1940, 1950 and 1970). 274 Ch. 4 Logic as the Universal Science I constants of the language to the universe of the model. Given an interpretation function, sentences can be assigned truth-values relative to the model. For example, for a sentence of the form P(a), the truth- definition has the following clause: M ~= P(a) if and only if I(a) I(P), i.e. a sentence of the form P(a) is true in a model M if and only if the interpretation of the constant a (a member of the universe) is a member of the interpretation of the unary predicate P (a subset of the universe). For complex sentences the truth-definition specifies the contribution of logical constants to the truth-value of sentences. For example, for conjunction the truth-definition has the following clause: M ~= P & Q if and only if M ~= P and M ~= Q, i.e., a sentence of the form P & Q is true in a model M if and only if P is true M and Q is true in M. [7] Given the semantics (truth-definition), we can further define: a schema is valid if and only if it is true under every interpretation (true in every model); a schema is a logical consequence of another schema if and only if every interpretation that makes the latter true also makes the former true. [8] The notion of schematization is a converse of interpretation; that a sentence can be schematized by a schema – is an instance of a schema – is to say that there is an interpretation under which the schema becomes the sentence.12

12 Goldfarb’s use of “sentence” calls for a comment. On the model- theoretic conception, sentences qua bearers of truth-value are set-theoretic entities. They result from assignments of interpretations to well-formed formulas. An example would be the pair , in which the first member is a syntactically specified formula, and the second member is a pair of interpretations, one for the predicate-letter, one for the individual constant. Such an entity bears little resemblance to sentences in the ordinary sense of that word (sentences as set-theoretic constructs are best thought of as models of natural language sentences; cf. here Shapiro (1998)). More importantly, they do not fit well with what Goldfarb says about interpretation and schematization (see points [8] and [9] above). Here it would be more natural to think of interpretation as replacement, i.e., as a procedure whereby, for example, the formula of the form P(a) is transformed into the sentence Ch. 4 Logic as the Universal Science I 275

[9] The logical properties of sentences are those of a schema of which the sentence is an instance (or, better, can be regarded as an instance for the purpose at hand); thus, a sentence is valid if and only if it can be schematized by a schema that is valid, and one sentence is a logical consequence of another if and only if they can be schematized by schemata of which the first is a consequence of the second. A notion of special importance is logical consequence. When a sentence S is said to be a logical consequence of a sentence R, this amounts to the following:

(A) There are schemata R* and S* such that, under some interpretation, R* yields R and S* yields S; and (B) under no interpretation is R* true and S* false.

This is (a special case of) the model-theoretic definition of logical consequence. It renders the notion of logical consequence sufficiently precise to allow the mathematical investigation of the notion. For instance, with the model-theoretic definition at hand, we may ask about the proof procedure of some formal system whether it is complete and sound.

“run(Timothy)” of a regimented fragment of a natural language. And we find that, in addition to the notion of interpretation qua assignment, Gold- farb (2001, p. 26) does mention the notion of interpretation qua replacement, which readily yields the two notions of interpretation and schematization as he describes them. Goldfarb’s usage of “sentence” best complies with a distinction between pure and applied logic (which he himself draws; id., p. 27); pure logic would be the province of schemata and their properties, whereas applied logic applies these schemata to the sentences (in some down-to-earth-sense) of some language – say a language in which a mathematical theory is formulated – to see whether they can be schematized by suitable schemata. 276 Ch. 4 Logic as the Universal Science I

4.2.2 Two Conceptions of Generality

Like any other science, logic strives for generality. As Goldfarb explains, the schematic conception secures this by the use of logical forms qua schematic representations. Logical forms “schematize away the particular subject matter of sentences” (2001, p. 27), and hence they deal with “what is common to and can be abstracted from different sentences” (id., p. 26). Such a view, however, is not foreign to the universalist conception, either (of course, Goldfarb is not unaware of this). Russell, for one, repeatedly used the language of “abstraction” in his attempts to explain what logic is about.13 Consider the following argument, which is of the kind that one tends to find in elementary text-books on logic:

(1) All men are mortal Peter is a man Peter is mortal

Any conception of logic recognizes that the logician’s primary business is not with such particular arguments as (1); these come within his purview only because logic can be applied to them: although it does not take a logician to tell us that (1) is valid, there is available a relatively simple logical technique, describable in different ways, that can be applied to (1) to show that it constitutes a valid argument. But arguments like (1) have another – philosophical – too; they can be used to introduce the subject-matter of logic. In particular, (1) enables us to make two elementary observations. Firstly, there is the observation, based on considerations which rely on an intuitive notion of substitution, that there are indefinitely many arguments that are relevantly similar to (1), or that (1) is representative of a certain kind of argument. Secondly, there is the observation that all arguments of this particular kind possess a certain logical property, namely validity. Putting

13 This is discussed in detail below, section 5.7. Ch. 4 Logic as the Universal Science I 277 these two observations together, everyone’s first idea is likely to be that, as far as (1) is concerned, its validity has nothing to do with its being about Peter (whoever he is) or about the two properties of being a man or being mortal. Rather it has something – perhaps everything – to do with the fact that (1) exhibits a certain pattern. This in turn can be made explicit by schematizing (1) or, to use more traditional terminology, by abstracting from its particular content. This procedure transforms (1) into

(2) All Fs are Gs x is F x is G

The recognition that particular arguments have underlying patterns constitutes just the beginning of the explanation of why (1) – or any other argument of this particular kind – is valid, and there are different ways to fill in the details. For our purposes, however, the important point is that any conception of logic, be it model-theoretic or universalist or something else, admits this much schematization, that it recognizes such patterns as (2), rather than particular arguments like (1), as being what logic is really about (this need not mean that arguments qua patterns are the only sorts of entities that are of interest to a logician). On any conception of what logic is, logic is, as we might put it, “minimally schematic” in the sense that its subject-matter is somehow related to such patterns as (2), the recognition of which – the recognition that there is a distinction to be drawn between (1) and (2) – constitutes the first step towards securing the generality of logic. Where the universalist and the model-theoretic conceptions differ is in their respective views on the patterns or schemata themselves and their relation to generality. For the model-theoretic logician, schemata are there to be interpreted. A simple illustration is provided by a formula of propositional logic: 278 Ch. 4 Logic as the Universal Science I

(3) (p & q) o p

On the schematic conception, (3) consists of schematic letters (p and q) as well as “logical constants” (& and o). The meaning of the constants is fixed (which is why they are constants), whereas the schematic letters receive content through interpretation. In this case, p and q are sentence-letters, and their interpretations are truth-values (they might also be suitable propositional entities, like declarative sentences, in which case we would interpret the schema by choosing suitable sentences to take the places indicated by the letters p and q (they are, indeed, often referred to as “placemarkers” or “placehold- ers”); in that way, however, one ends up doing applied and not pure logic). The assignment of truth-values to p and q (the primitive symbols or atomic formulas) puts us in a position to interpret the entire complex formula (3), i.e., calculate its truth-value. On the universalist conception, by contrast, p and q are understood as variables rather than schematic letters to which interpretations can be assigned. The schema is given content not by interpreting it but through quantification.14 Keeping in mind that logic strives for generality, (3) gives rise to its universal closure:

(4) ( p)( q)((p & q) o p)

How is (4) to be understood, according to the universalist? There are two strategies available here, one Fregean, the other Russellian. Ac- cording to the Fregean strategy, variables are segregated into types, so that, for example, (4) is a generalization over all propositional entities (be they sentences, statements, propositions, or something else), and the unrestricted generality that is typical of the universalist conception lies in the fact that variables range over all entities of the suitable

14 The universalist need not deny that p and q can be given interpretations, but the procedure is not exactly the same as the model-theoretic logician’s interpretation. The results of interpretation in this sense, furthermore, belong at most to an application of logic, and they not germane to pure logic. Ch. 4 Logic as the Universal Science I 279 type. By contrast, the early Russell thought that allowing even this much restriction betrays the true universality that is the characteristic of logic. His strategy is to add to the original formula, “(p & q) o p”, a condition specifying, as it were, its “intended interpretation”. Since “(p & q) o p” applies to propositions, the condition must be one that distinguishes propositions from all other entities. Russell uses the notion of implication for this purpose, as he holds that propositions are the only entities that stand in the relation of implication: “p o p”, though it may not in fact state that p is a proposition, at least expresses a condition which is satisfied by propositions only. (4) is thus transformed into:

(5) ( p)( q)(p o p and q o q) o ((p & q) o p)), that is, if p and q are propositions, then p and q together imply p. In this way, the generality requirement is fulfilled: in (5) variables p and q range over everything, and the proposition is true for non- propositional values of p and q, since they fail to satisfy the antecedent.15 For another example, consider following schema:

(6) (FxȺ Gx) Ⱥ ((GxȺ Hx) Ⱥ (FxȺ Hx))

On the schematic conception, ‘F’, ‘G’ and ‘H’ are predicate-letters, i.e., expressions that receive suitable entities as their interpretations (sets over the domain of interpretation, or possibly, predicates as linguistic expressions). As for the universalist, he understands them, again, as quantifiable variables:

15 This is the approach that Russell adopted in the Principles. As will be explained below (see fn. 110), this is not in fact the only approach available to the universalist who is not attracted to the Fregean approach; with sufficient ingenuity the spirit of the Russellian approach can be retained while avoiding the complicated way of stating the propositions of logic that is used in the Principles. 280 Ch. 4 Logic as the Universal Science I

(7) ( F)( G)( H)[( x)(FxȺ Gx) Ⱥ (( x)(GxȺ Hx) Ⱥ ( x)(FxȺ Gx))]

As in (5), the variables occurring in the formula can be taken in two different ways. The Fregean view is that the variables bound by “( F)”, “( G)” and “( H)” “are to be understood” as ranging over suitable predicable entities (properties, concepts, functions, etc.); this could be called the strategy of implicit condition – implicit, because the condition is not attached to the formula, but is shown by the use of upper-case letters. The alternative, Russellian strategy is to modify the formula by adding to it a clause specifying what kinds of entities satisfy the formula; this could be called the strategy of explicit condition. The strategy of explicit condition is apt to give rise to certain difficulties having to do with the notorious issue of propositional unity (and the Fregean or implicit strategy was designed to overcome just this problem). Further discussion of these difficulties, however, is best deferred until we come to a more detailed presentation of Rus- sell’s version of the universalist conception16

4.2.3 The Technical Core of the Universalist Conception

Formulas (5) and (7) enable us to make several observations concerning the technical core of the universalist conception. In each case the contrast with the schematic conception is as evident as it is immediate.17 Firstly, (5) and (7) are true and general statements. It is therefore appropriate to call them laws of logic. Whatever the term, they are the sort of statements that a logician asserts.

16 See section 4.4.6 below. 17 Since the following points are repeated, in more or less detail, in every paper on the topic, I will for the most part omit references to secondary literature. Two clear presentations are Goldfarb (2001) and Kemp (1995). Ch. 4 Logic as the Universal Science I 281

Secondly, the generality of the laws of logic is not different in kind from the generality that pertains, say, to physics or geometry. The only difference is that the laws of logic are more general than the laws of the special sciences. Indeed, the logician’s assertions are the su- premely general statements or the most general statements that can be made: in addition to variables, the laws of logic require for their expression only such terms that feature in all discourse, or are needed in thought and reasoning about any subject-matter or topic whatsoever (Russell used to call them “logical constants” and the term has caught on).18 On the schematic conception, by contrast, the logician’s assertions are not truths of logic (logical truths) like (5) or (7). Rather, the logician merely specifies these truths, as statements possessing a certain form. In other words, the logician’s assertions are generalizations over schemata. Hence, we can say that, whereas for the universalist the generality of logic is substantive, it is merely schematic for an advocate of the schematic conception. On the latter view, the generality of logic resides in the fact that a formalism can be given different interpretations, whereas the universalist logician’s assertions are truths about everything that are made by dint of the topic-neutral vocabulary of logic. Thirdly, the association of the laws of logic with scientific laws is still further enforced by the fact that (5) and (7) express plain truths. That is to say, their truth is not different from the truth that characterizes the general statements belonging to special sciences. This means, as a first approximation, that they are made true by facts about the actual world that differ from the facts of the special sciences only by being more abstract or general.19 By contrast, the schematic conception uses, not plain truth, but

18 Issues relating to this notion of generality will be discussed in section 5.7. 19 Much later, Russell gave a well-known expression to this view: “logic is concerned with the real world just as truly as zoology, though with its more abstract and general features” (1919, p. 169). 282 Ch. 4 Logic as the Universal Science I the relational notion of “true-in-a-model” or “true-under-an- interpretation”. It is natural to assume, however, that the two notions must be closely related: since “true under an interpretation” means “true, when interpretations have been assigned”, it seems that the model-theorist’s notion of relative truth is parasitic upon plain truth (Hodges 2005).20 Fourthly, the two conceptions differ in their treatment of truth- predicate. The schematic conception uses a truth-predicate in its definitions of validity and logical consequence, which talk of truth under all interpretations of schemata. And since there are infinitely many sentences to which the predicate applies, it cannot be eliminated by dint of the disquotational schema (“‘p’ is true if and only if p”). The universalist conception, on the other hand, has no comparable role for the truth-predicate: it has no role to play either in the formulation or application of the laws of logic (Goldfarb 2001, p. 30) Fifthly, the sense in which logic is applied is different in the two cases. For the universalist, application amounts to instantiation, whereas the other view has it that logic is applied by considering some language and seeing whether its sentences can be fruitfully regarded as interpretations of schemata. Sixthly, and this point sums up the points previously made, the relation that logic bears to reality is different on the two conceptions. For the universalist, logic is straightforwardly about the world: logic, to

20 In reality, however, it depends on what it means to “assign interpretations”. John Etchemendy (1983, 1990, Ch. 2) points out that the models of model-theory can be looked upon in two radically different ways, one giving rise to “interpretational”, the other to “representational” semantics. In interpretational semantics, we change the interpretations of some expressions to see how the resulting sentences behave with respect to ordinary or plain truth. In representational semantics, sentences retain their meanings, but we imagine different counterfactual situations to see how the sentences behave with respect to these counterfactual situations. On the former view, “plain truth” is the fundamental notion, whereas the latter view sees plain truth (“true in the actual world/situation”) as a special case of the more general notion of “true in a world/situation”. Ch. 4 Logic as the Universal Science I 283 borrow a phrase from Gary Kemp, “stands in the same first-order relation to reality as any other science” (1996, p. 170). According to the schematic conception, by contrast, logic is a metadiscipline. This means that logic is not about extra-linguistic reality, but about statements and inferences that are themselves about reality. When Michael Dummett argues that “[r]eality cannot be said to obey a law of logic; it is our thinking about reality that obeys such a law or flouts it” (1991, p. 2), this view, though applicable to much of contemporary thinking about logic, is simply wrong, if applied to the universalist conception.21

4.3 The Philosophical Implications of the Universalist Conception

4.3.1 “No Metaperspective”

Our next task is to spell out the philosophical implications of the universalist conception. Here we are entering an area that is more controversial. In the previous section we saw how, on the universalist conception, logic stands firmly in a first-order relation to reality. This means that the statements of logic have content and a truth-value on their own, independently of any external interpretation. Conse- quently, a meta-perspective – a standpoint from which we can look at and survey logical formulas and make judgments about their properties – is not germane to this way of thinking about logic. It is this notion of “possessing content on its own” that the term “language” is calculated to capture in the first place; the contrast, it will be recalled, is with a conception of logic as calculus, and here the use of the term “calculus” is meant to emphasize the importance for logical theorizing of semantics (the possibility of reinterpretation). According to the van Heijenoort interpretation, the universalist does not regard logic simply as a language; rather, it is to be seen as

21 As Ricketts (1996, p. 123) and Goldfarb (2001, p. 29) point out. 284 Ch. 4 Logic as the Universal Science I the universal medium. Universality, then, implies more than just the view that logic is a language with content. For logic is universal also – indeed, primarily – in the sense that it is unique: there is but one logic, and since it is a natural assumption that all (deductive) reasoning is logical in character, uniqueness virtually implies that all reasoning, simply in virtue of being reasoning, falls within the purview of logic.22, 23 Combining the view that logical formulas have fixed content or fixed sense with the further assumption that logic is unique, we seem to end up with the conclusion that the idea of a meta-perspective (metalogic, metalanguage) is not only not germane to logic but simply falls away and becomes impossible. This, at any rate, is how scholars in the van Heijenoort tradition tend to express the point. The tradition is thus characterised by the following piece of reasoning, which it attributes to the advocates of the conception of logic as language:

because logic is universal, it encompasses all (deductive) reasoning or all thought; therefore there is no external standpoint from which

22 A different way of putting the point, and one that emphasizes the language-aspect of the universalist conception of logic, would be to say that, for a universalist, logic is not so much a tool for reasoning as it is a “universal language in which everything can be expressed” (Hintikka 1997, p. 22). Ei- ther way, the universality of logic lies in its “inescapability” (Hintikka and Hintikka 1986, p. 2) or “exclusiveness” (1997, p. 22). Insofar as we are en- gaged in thought and talk – and this includes (deductive) reasoning as a special case – we must use the one and only logic there is. 23 Strictly speaking, the assumption of uniqueness runs together two distinct theses. Firstly, there is the thesis that there is but one logic. Secondly, there is the thesis that logic covers all thought and all (deductive) reasoning. These two theses are independent of each other. For one could accept the former while rejecting the latter. For example, it is a reasonable assumption that Kant thought that there is but one logic, namely, “traditional logic”, but he also thought, as we have seen, that this “one and only logic” does not encompass all deductive movements of thought. A believer in the universality of logic, by contrasts, holds that the one true logic is all-encompassing in that it exhausts these movements. Ch. 4 Logic as the Universal Science I 285

we can raise metasystematic questions.

Here are three quotations that give a very clear expression to this line of thought:

Frege’s and Russell’s systems are meant to provide a universal language: a framework inside which all rational discourse proceeds. Thus there can be no position outside the system from which to assess it. The laws they derive are general laws with fixed sense: questions of interpretation and disinterpretation cannot arise. All this distinguishes their conception of logic from that more common today, which relies on schematization and interpretation, and defines logical truth by reference to schemata. (...) Frege and Russell can have no notion of “interpretation,” or of “semantics”. The text surrounding their formulas is at best heuristic, aimed at initiating their audience into their languages. Moreover, the logic they practice aims only at issuing general truths in this language. In particular, it does not issue metastatements of the form “X is a logical truth” or “X implies Y.” Logic, for them, does not talk of the forms of judgment. (Goldfarb 1982, p. 694)

For Frege, and then for Russell and Whitehead, logic was universal: within each explicit formulation of logic all deductive reasoning, including all of classical analysis and much of Cantorian set theory, was to be formalized. Hence not only was pure quantification theory never at the center of their attention, but metasystematic questions as such, for example the question of completeness, could not be meaningfully raised. We can give different formulations of logic, formulations that differ with respect to what logical constants are taken as primitive or what formulas are taken as formal axioms, but we have no vantage point from which we can survey a given formalism as a whole, let alone look at logic whole. (Dreben and van Heijenoort 1986, p. 44)

The fact that Russell does not see logic as something on which one can take a meta-theoretical perspective thus constitutes a crucial difference between his conception of logic and the model-theoretic conception. Logic, for Russell, is a systematization of reasoning in general, or reasoning as such. If we have a correct systematization, it will comprehend all correct principles of reasoning. Given such a conception of logic there can be no external perspective. Any reasoning will, simply in virtue of be- 286 Ch. 4 Logic as the Universal Science I

ing reasoning, fall within logic; any proposition that we might wish to advance is subject to the rules of logic. This is perhaps a natural, if naïve, way of thinking about logic. (Hylton 1990a, p. 203)

It should be pointed out that the conclusion is not always stated in such strong terms. At least on occasion, the advocates of the van Heijenoort interpretation use more cautious formulations. For instance, Hintikka and Hintikka (1986, p. 2) write: “it is important to realize that the thesis of language as the universal medium implies primarily the inexpressibility of semantics rather than the impossibility of semantics, in the sense that a believer in language as the universal medium can nevertheless have many and sharp ideas about language-world connections, which are the subject of semantics. How- ever, these relations are inexpressible if one believes in the view of language as the universal medium”. They use Frege’s semantics as an illustration. As they see it, Frege’s views on semantics amounts to much more than just the sense-reference distinction – a device intended primarily to deal with oblique contexts, according to the Hin- tikkas. In addition to that, Frege had a “definite and articulated set of ideas about the semantics of ordinary extensional language, including definitions of propositional connectives, the meanings of quantifiers, etc. However, in spite of being articulated, this set of semantic ideas does not really amount to a theory, for Frege did not believe in the proper expressibility of such ideas, and hence he regarded them as “indirect explanations” (ibid.)24 Nevertheless, the formulation that I will examine in the present work is one that speaks about impossibility, rather than something else, as this appears to be the most common formulation. Summing up the discussion so far, the gist of the universalist thesis is captured by the combination of the following two theses:

(UL-1) The laws of logic have fixed content/sense.

24 This point has also been made by several other scholars. The most detailed study of this issue, as it applies to Frege, is Ricketts (1997). Ch. 4 Logic as the Universal Science I 287

(UL-2) Logic is unique and all-encompassing.

The scholars in the van Heijenoort tradition typically endorse the following view:

x Acceptance of the universality of logic, i.e., of (UL-1) and (UL- 2), implies a sceptical attitude towards semantic notions and, more generally, semantic or model-theoretic ways of thinking: they are either illegitimate or, at the very least, they are not germane to logic, belonging at best to what could be called its propaedeutics.

4.3.2. Two Senses of “Interpretation”

The “anti-semanticism”, as we may call it, that characterizes the universalist conception has far-reaching philosophical implications for one’s understanding of logical theory and for a number of related logical concepts; this is of course the reason why the advocates of the van Heijenoort interpretation argue that the distinction between the two conceptions of logic (or language) is a useful interpretative tool. The systematic spelling-out of these consequences, however, is complicated by an interpretative issue that I have ignored so far. As I have presented it, the model-theoretic conception is one whose proponents look at the formulas of logic in a certain way (as schemata) and, consequently, make use of a certain kind of argumentation (conceptualisations and techniques which are recognisable, by hindsight or by the proponents themselves, as model-theoretic). This is of course an eminently plausible approach; after all, the interpretative framework suggested by van Heijenoort is supposed to throw light on the development of modern logic, and one of the more significant characteristics of this development is, precisely, the gradual emergence of the concept of model in the technical sense of that term. It is this technical sense, related to logical theory on some suffi- 288 Ch. 4 Logic as the Universal Science I ciently narrow construal of that term, that figures in much (indeed, most) of the secondary literature that has been written about the van Heijenoort interpretation. For the sake of completeness we must nevertheless note that this is not the only perspective available. For model-theoretic conceptualisations can be understood in a more philosophical sense, a sense that is recognisably different, though of course not unrelated, to the technical one. To see the difference between the two perspectives, consider the following two lists, in which the two conceptions of logic are compared on a number of specific points; the list is adapted from Kusch (1989, pp. 6-7) and Hintikka (1997, pp. 227-8):

Universality of logic/language The model-theoretic tradition

1. Interpretation cannot be varied. 1. Interpretation can be varied. 2. Model theory is impossible (or 2. Model theory is possible (and irrelevant). important). 3. Only one world can be talked 3. Many worlds can be talked about. about. 4. One domain of quantification 4. Ranges of fully analysed quan- in the last analysis. tifiers can be different. 5. Logical truths are about this 5. Logical truth as truth in all (the actual) world. possible worlds. 6. (Genuine) metalanguage is 6. Metalanguage is possible and impossible. legitimate. 7. Truth as correspondence is 7. Truth as correspondence is a unexplainable/unintelligible. legitimate notion.

The basic observation about these lists is that the key term “interpretation” can be understood in more than one way. We have in fact already met one instance of this ambiguity. Above it was mentioned that the schematic conception uses the notion of relative truth (true- in-a-model), whereas the universalist notion of truth is the notion of plain truth (or absolute truth; see above, footnote 17). It was then pointed out – and this is the basic observation – that one’s answer to Ch. 4 Logic as the Universal Science I 289 the question as to which of these two notions is, conceptually speaking, more basic depends upon how one understands the “assignment of interpretations”. To put the point another way, the answer to the question depends upon what one takes models to be models of. If models are there to give meanings-of-sorts to the expressions of language, i.e., to certain syntactically specified objects, then plain truth emerges as the more fundamental concept. This is the role of models or interpretations on the schematic conception. That conception starts with sentences as syntactically specified strings of symbols. Evidently, such entities can have truth-values only after they are interpreted or assigned meanings(-of-sorts). On the schematic conception, however, this giving of meaning must be taken in a very thin or technical sense. This sense should not be conflated with some more robust sense. In the technical sense, one has given meanings to expressions – string of symbols – as soon as one has specified a domain of objects and related, with the help of an interpretation function, the strings to these objects and sets formed from them. This procedure belongs within what John Etchemendy has called interpretational semantics (see, again, footnote 17). The representational approach is very different. There it is assumed that expressions already possess meanings (in something like the customary sense of that expression), and these expressions-with- meanings are then related to models qua situations which the language in question can be used to speak about. This is the basic idea of possible worlds semantics (modal semantics), where meanings – intensions, as they are customarily called – are conceptualised as functions from worlds (plus, possibly, other indexes) to extensions. Model theory in the technical sense and modal semantics serve very different purposes. Model-theoretic techniques are used to survey formal theories for a number of logical properties, including such global properties as deductive completeness, and such local properties as the consistency and independence of axioms. Modal semantics, on the other hand, is intended as a tool for philosophical analysis, i.e., as a means that is capable of throwing light on a number of philosophi- 290 Ch. 4 Logic as the Universal Science I cally important notions like necessity, possibility, knowledge, belief, permission, obligation, etc.25 Returning now to the list, it is clear that its originators have looked at things, by and large, from the standpoint of modal semantics. From this standpoint it is tempting to characterize the universalist conception simply as the denial of the possibility of modal semantics. Very roughly, such a denial amounts to holding that since the interpretation of our canonical language has been given once and for all, there is but one language and this language speaks of one world only.26 Hintikka has argued that essentially this picture underlies Frege’s and Russell’s logical theories and their semantics. As he sees it, the crux of the matter is the following line of thought:

The starting-point of this development [sc. the development of model- theory] was the conception of Frege and Russell which included the uni-

25 Cf. here Hintikka (1975). On the face of it, this distinction between the two disciplines is blurred by the fact that model theory in the technical sense is at least occasionally thought of as capable of supplying “analyses” or “explications” of such notions as truth, logical consequence and logical truth. It could be argued, however, that this only reflects the existence of two different notions of analysis. And there is room here for an argument to the effect that the step from models in the technical sense to worlds or situations in some realistic sense is obligatory if models are to do some real work in philosophers’ hands. For example, Hintikka (ibid.) argues that Carnap’s semantic approach to modality was vitiated by his failure to appreciate just this point: “possible worlds were not thought of by Carnap as the real-life situations in which a speaker might possibly find himself, but as any old configurations – perhaps even linguistic – exemplifying appropriate structures” (id., p. 223). It was this myopia, according to Hintikka, that foreclosed, on Carnap’s part, the possibility of several potentially fruitful applications of his semantic machinery. Clearly, any such argument must be grounded in a view as to what kind of work the semanticist’s models are supposed to; for some steps in this direction, see Hintikka (1969b, Ch. 1). 26 It is tempting to formulate this conclusion in the ontological idiom, saying that there is only one world. In the present context, however, the locution “there is” turns out to mean no more and no less than “one can speak of”. Ch. 4 Logic as the Universal Science I 291

versality of language in two different ways. They took the impossibility of alternative interpretations of our language to amount to a kind of one- world doctrine. In order to speak of alternatives to this world of ours, we would have to (they thought) alter the interpretation of our language, for actually it speaks of course of the actual world. [...] Hence it does not make any sense to speak of “models” other than the actual world on which our language has been interpreted. Since we cannot discuss what would happen if the interpretation were changed, we cannot really speak of alternatives to the actual world. (1997, pp. 25-6)

Whether this is an accurate picture of Frege’s and Russell’s reasoning is a question on which I shall not try to decide here. For now the important point is that there is, from the universalist point of view, nothing inherently objectionable in the idea of a multiplicity of interpretations, insofar as this is understood in something like the sense licensed by interpretational semantics.27 Approaching the matter from the other direction, we can formulate the point by saying that the use of conceptualisations or techniques that are based on the availability of a multiplicity of interpretations does not by itself lead to an acceptance of modal semantics. These considerations suggest that a verdict on the universalist thesis and its implications presupposes a clear recognition that such terms as “interpretation” and “model” have a multiplicity of uses. Like most of the notions that we tend to take more or less for granted, the concept of interpretation, as it figures in logical theory, has a complex history, and the question of whether or not a past logician “accepts” or “rejects” the possibility of something worthy of the name “reinterpretation” and what prima facie implications the acceptance or rejection might have is one that cannot be answered offhand (of course, this same remark applies to most of the other terms that figure in the contrast between the two conceptions).28 In practice,

27 I shall return to this point below, in section 4.5.4, where it is argued in detail. 28 I do not mean to suggest that Hintikka is unclear on these points. On the contrary, he is well aware that model theory in the restricted sense is one 292 Ch. 4 Logic as the Universal Science I though, scholars have, as a rule, worked with the schematic conception of interpretation, and this assumption will also be made in the present work. That is, I shall concentrate on the universalist thesis and its implications for logical theory in some relatively narrow sense of that term.

4.3.3 A Flowchart

The following flowchart is intended to capture the essential content of the universalist conception of logic, as the van Heijenoort interpretation represents it: thing, and questions concerning the feasibility of model-theoretic conceptualizations on some larger scale (“modal semantics”) something quite different. For example, Quine has admitted, according to Hintikka, the possibility of model-theoretical conceptualizations “in a small scale” (1997, p. 217). This, however, does not make Quine a proponent of the calculus view, on Hintikka’s understanding of what this view amounts to. For he has consistently refused model theory any role in the semantics of natural language. That he is committed to “universalism” is shown, among other things, by “the emphasis on one’s home language as the medium of all theorizing and philosophizing about language of all language teaching” (id., p. 216), and the related view that observable (linguistic) behaviour supplies all there is to meaning. That Hintikka looks at the distinction between the universalist and model-theoretic conceptions from the standpoint of modal semantics is a natural consequence of the fact that he is interested in more than just an elucidation of the development of modern logic or logical theory in some relatively “narrow sense”. His use of the interpretative framework amounts to an attempt to find a perspective from which, to put it rather loosely, analytic philosophy in its entirety can be surveyed and assessed. Given the unquestionable importance that the so-called linguistic turn had for analytic philosophy, Hintikka’s interpretative strategy seems eminently reasonable. Of course, it can be challenged, since it is not indisputable that modal semantics is really the great watershed in twentieth-century (analytic) philosophy that Hintikka thinks it is. Ch. 4 Logic as the Universal Science I 293

Figure 4.1 The universalist conception of logic according to the van Heijenoort interpretation

A number of comments are needed. Firstly, the core of the universalist conception is given by the two uppermost boxes (theses (UL-1) and (UL-2), which we have already met). Thus, the conception combines a certain view of the formulas of logic (that they have fixed content or sense) with the further assumptions that logic is unique and all- encompassing. These two (or three) theses have a number of prima facie implications which have a common denominator, to wit, anti-semanticism. 294 Ch. 4 Logic as the Universal Science I

Secondly, in drawing the flowchart, I have assumed that the universalist thesis amounts to the impossibility of semantics, rather than the “not germane to logic” attitude. As I pointed out above, it is the impossibility thesis that is more commonly discussed in the secondary literature and will be assumed here as well. Thirdly, it is evident that the inference of “semantic relations cannot be varied” and “metalanguage is impossible” from the core theses of the universalist conception are crucial for the van Heijenoort interpretation. It must be said that this inference is not self-evident. I have already commented briefly on the notion interpretation, pointing out that its content can be construed in more than one way, which in turn makes its presuppositions and implications less than transparent. Analogous remarks apply to the notion of metalanguage or metaperspective. There is, to begin with, a considerable risk of anachronism here. This has been emphasized by Jamie Tappenden (1997, pp. 222-3), and I concur with his conclusions. The scholars in the van Heijenoort tradition seem to be pretty much unanimous in thinking that the universalist conception excludes an external perspective on logic. For example, we have seen Hylton attributing to Russell the following reasoning: since logic encompasses all reasoning, there can be no external perspective; therefore, there cannot be a meta-theoretical perspective on logic. Tappenden points out, as against attributions of this kind, that it is highly unlikely that someone like Frege – and, we may add, Russell – would have thought that semantic or other meta-theoretic investigations require something that deserves to be called an external perspective. For us, such a view may be a matter of course, given the results that Tarski established in the 1930s: we have come to accept, more or less, that the semantics for a language must be given in a metalanguage that is in certain ways stronger than the language for which semantics is being given.29 But the view itself that semantic theorizing

29 Whether this conclusion is correct is something that will not be discussed here. Ch. 4 Logic as the Universal Science I 295 requires an external standpoint may not be particularly natural; rather, it is a conclusion to which one is forced for reasons that are theoretical in nature.30 Thus, it could be argued that the prima facie implication of the universalist conception is the exact opposite of what the advocates of the “no metatheory” interpretation assume: if logic really is universal and includes absolutely everything, the conclusion to be drawn from this is that this everything should include logic as well.31 If one thinks otherwise, arguing that logic is universal in the sense that its scope includes absolutely everything, and concludes that for this reason there can be no theory that is about logic, one must have extra reasons, or reasons that are independent of the assumption of universality. Considerations of what is natural certainly suggest that the universality assumption is in itself insufficient to undermine the possibility of meta-theory. It may well be that universalists have been inclined to the “no semantic meta-theory” conclusion. But if the above line of reasoning is persuasive – and I think it is – this is something that needs to be established separately in each case. Quite evidently, we cannot assume without further ado that the reasoning by which Tarski was led to his conclusions was operative in the case of someone like Frege or Rus- sell. And the situation is not changed even if we substitute some other name for “Tarski”. Wittgenstein’s Tractatus is in many ways the paradigmatic example of the universalist conception of logic and language. And Wittgenstein did think that a genuine metaperspective would require that we accomplish something that is impossible, namely, that we station ourselves somewhere outside logic, language and the world. Again, however, the same point applies: even if we are inclined to see Wittgenstein as the only really consistent universalist, and think that his views represent the universalist conception in its pure form, we cannot conclude from this that his views and the reasoning behind them were shared by other advocates of the universalist conception. In reconstructing their views, for example, those of

30 Tarski’s case could be used as an illustration here. 31 Cf. Tappenden (1997, p. 223). 296 Ch. 4 Logic as the Universal Science I

Russell, such elementary observations should be kept in mind. Fourthly, the universality thesis is meant to have consequences for semantics only. What is impossible, according to the universalist, is semantics as a systematic enterprise (or model-theoretic conceptualisations): such an enterprise presupposes the possibility of disinterpretation and reinterpretation of (non-logical) vocabulary, and this, according to the interpretation, is a no-no for a believer in the universality of logic.32 This impossibility, however, does not preclude the possibility of speaking, and even framing a theory, of “the words and other symbols of the language, abstracted from their semantical function” (Hintikka and Hintikka 1986, p. 171). What is possible for a believer in the universality of logic, in other words, is a purely syntactic approach to logic.33

32 It goes without saying that scholars in the van Heijenoort tradition may understand the implications of the universality thesis in different ways. It is therefore strictly speaking not correct to claim, as I do in the text, that “the universality thesis is meant to have consequences for semantics [...] only”; different scholars may mean, and are free to mean, different things by the thesis. A more cautious formulation would be to say that there is a prima facie implication from the universalist conception to the “inequality” between syntax, which is a legitimate discipline, and semantics, which is not. 33 According to Hintikka and Hintikka (1986, pp. 9-10), this is precisely what we find in Frege and the early Wittgenstein. Both were firmly convinced that logic has a foundation in a system of meaning relations. Never- theless, they also believed that this foundation cannot be made an object of systematic study, or a theory, this being a consequence of the fact that they were both believers in the universality of logic or “language as the universal medium”. What was left for them, therefore, was the study logic as a “formalism” or purely syntactic object. Thus, Frege was the first to develop a complete formalization of first-order logic, in fact the first to conceive of logic as a formal system in anything like our present-day sense of this term, and yet he was consistently and relentlessly hostile to all “formalists”. Witt- genstein in turn was largely responsible for the introduction of the idea of “logical syntax”. (Even if one concurs with what the Hintikkas have to say about Frege, one should exercise some care in applying this idea to the Trac- tatus. For it does not seem that Wittgenstein saw any relevant difference in this respect between semantics and what he recognized as syntax: both be- Ch. 4 Logic as the Universal Science I 297

It is nevertheless arguable that the universalist conception does have implications for one’s understanding of issues that Frege (or we) would perceive as belonging to syntax. To see matters in Frege’s way presupposes that one is clear about the distinction between syntax and semantics. And it is not far-fetched to think that Frege may have been uncommon among the logicians of his day in perceiving syntax as an autonomous discipline and also in drawing a sharp distinction between object-language (judgments that constitute his concept- script) and metalanguage (judgments about the syntax and semantics of concept-script).34 At any rate, it is a familiar observation that Rus- sell, for one, was less conscious than Frege of such subtleties.35 Con- sequently, it is tempting to trace confusions over such distinctions – or less derisively, failures to conceive these matters in terms that we tend to take for granted – to the influence of the universalist conception. Fifthly, the flowchart does not include each and every of the putative consequences of the universalist conception. One set of implications that I have only indicated is formed by the metasystematic questions, of which the question of the definability of truth is but one example. Another implication which I have not mentioned by name is worth mentioning here, since it is of considerable importance in the case of such universalists as Frege and Russell. The status that one gives to model theory and model-theoretic reasoning has obvious bearings on one’s views on mathematical theories. This is best illustrated by the famous Frege-Hilbert exchange, which is probably the clearest actual example of a clash between a universalist and a model-theorist. To what extent the early Russell’s views on mathematical theories long to the domain of “showing,” rather than “saying”; for the early Witt- genstein’s conception of syntax, see Emiliani (1999). 34 Juliet Floyd (1998) has argued, however, that even the case of Frege may be less clear in this respect than is traditionally thought. 35 As Gödel, for one, pointed out; see Gödel (1944, p. 126). Russell’s is not the only such case. The name of Peano is often cited as another example; see Zaitsev (1994) and below, section 4.5.2.2.1 for further discussion. 298 Ch. 4 Logic as the Universal Science I were informed by the universalist conception is a question that deserves to be addressed in some detail.36 Sixthly, it would be very satisfactory, from an interpretative point of view, if the theses that constitute the philosophical core of the universalist conception, rather than simply assumed as the starting- point of the interpretation, could somehow be motivated. It is, I think, quite natural to insist on an explanation here: why should anyone want to advocate such apparently rather extreme views on a number of key issues in logical theory?37 In Russell’s case, it is really not too difficult to come up with a plausible explanatory hypothesis, namely his notion of proposition. As is well-known, a radically non-standard notion of proposition has an important role to play in the early Russell’s philosophy. This notion combines features from more orthodox theories of propositions – Russellian propositions are truth-bearers – with features that make these entities look more like states of affairs – Russellian propositions are complexes of worldly entities. From this account of propositions, it may be argued, there is a relatively straightforward path to the universalist conception of logic. Simply put, the basic point is. The universalist conception centres around the rejection of semantic notions. If, now, propositions are non-linguistic – more generally, non- representational – entities, the conclusion lies at hand, minimally, that the semantic perspective cannot be germane to logic. It is this line of thought that will be developed and examined below. But there is another reason for being interested in the Russellian proposition. For this notion is really the centrepiece of the early Rus- sell’s conception of logic, and its relevance is therefore not restricted to what it implies about the van Heijenoort interpretation. It is for this reason that I will first examine the notion of Russellian proposition on its own as well as the uses to which he puts it. Here the pri-

36 This question is considered in section 4.6. 37 The question becomes particularly stressing, I believe, if one holds that the real ultimate premise for is the somewhat elusive doctrine of “ineffability of semantics”; see section 4.5.3. Ch. 4 Logic as the Universal Science I 299 mary interpretative task will be to identify the complex problems to which the introduction of the notion was calculated to give answers. Once we have formed an adequate picture of the relevant issues, we can profitably return to the many interpretative issues that emerge, when the van Heijenoort interpretation is applied to the early Russell.

4.4 Russell’s Notion of Proposition

4.4.1 Preliminary Remarks

At the beginning of his survey of Meinong’s theory of objects, Russell enumerates four theses which he says he has been led to accept by “Mr. G. E. Moore”. These four theses can be formulated as follows:

(1) Every mental act has an object which is distinct from the act and is, in general, an extra-mental entity; (2) perception has as its object an existential proposition; (3) truth and falsehood apply not to mental acts (beliefs, judgments) but to their objects, i.e. propositions; (4) whether or not they exist, the objects of mental acts are or have being; this ontological status is independent of their being objects of thought (Russell 1904a, p. 431).

Russell says that these views are “generally rejected” and adds the somewhat cautious note that each of them can be supported by arguments which “deserve at least a refutation” (ibid.) This might suggest that the four theses were put forward merely as hypotheses with no firm commitment, on Russell’s part, as to their truth or otherwise. But the fact is, of course, that Russell was firmly committed to them and that he regarded them as the keys to all sound philosophizing. The import of (1)-(4) is highly non-trivial. With appropriate elaboration and substantiation, they comprise at least (i) a theory of intentionality or thought, (ii) a theory of meaning, (iii) a philosophy of logic, 300 Ch. 4 Logic as the Universal Science I including a theory of truth, and (iv) a general ontology. Within each of these, the notion of proposition is given a fundamental explanatory role. As regards (i), it suffices here to emphasize the fact that Russell’s position is fundamentally at variance with or inimical to any kind of “representationalism”. The intentionality or “aboutness” of thought is to be explained along the lines of what is known as the act-object model, that is to say, without assuming any intervening entities. The most striking expression of this idea is to be found, perhaps, in Moore’s The Refutation of Idealism:

There is [...] no question of how we are to ‘get outside the circle of our own ideas and sensations’. Merely to have a sensation is already to be outside that circle. It is to know something which is as truly and really not a part of my experience, as anything which I can ever know. (Moore 1903b, p. 451; all italics in the original)

As the quotation indicates, the idea underlying the act-object model, at least as this was understood by Moore and Russell, is to close the gap between thought and the world.38 This means in particular that the correctness or otherwise of a thought is not to be explained by referring to

38 Russell makes much the same point in a letter to Frege, written on 12 December 1904. In an earlier letter Frege had argued that Mont Blanc, the mountain itself, is not a constituent part of what is asserted in the sentence (Satz) “Mont Blanc is over 4000 metres high”, and that, consequently, we must distinguish between the Sinn of a Satz (that which is asserted), and its Bedeutung. Russell wrote the following as a reply: “Concerning Sinn and Bedeu- tung, I cannot see but difficulties which I cannot overcome. I believe that in spite of all its snowfields Mont Blanc itself is a component part of what is asserted in the Satz ‘Mont Blanc is more than 4000 metres high’. We do not assert the thought, for this is a private psychological matter: we assert the object of the thought, and this is, to my mind, a certain complex (an objectiver satz, one might say) in which Mont Blanc is itself a component part. If we do not admit this, then we get the conclusion that we know nothing at all about Mont Blanc” (Frege 1980, p. 169). I have followed Hylton (1990a, p. 172) in leaving the German “Satz” untranslated, so as not to obscure Russell’s point. Ch. 4 Logic as the Universal Science I 301 anything that is external to that thought. Suppose, to illustrate, that I entertain the thought that Bacon wrote Shakespeare’s plays. The condition that supplies the criterion for the correctness of this thought is in no way separable from the thought itself. If the thought is correct, i.e., true, then it is a fact that Bacon wrote Shakespeare’s plays, but this adds nothing to the characterization of my thought as true; facts simply are true thoughts, or in Russell’s preferred terminology, true propositions. The act-object model of intentionality carries over into the theory of meaning. There it corresponds, in a well-known manner, with two- tiered analyses of meaning:

Act Object Expression Meaning-qua-referent

In other words, just like our mental acts are directly related to their objects, linguistic expressions are directly related to their meanings. And since these meanings are worldly entities, they are appropriately called referents (this is the ‘Fido’- Fido -theory of meaning). From Russell’s standpoint, the most important thing about linguistic meaning is that it is not important:

To have meaning, it seems to me, is a notion confusedly compounded of logical and psychological elements. Words all have meaning, in the simple sense that they are symbols which stand for something other than themselves. But a proposition, unless it happens to be linguistic, does not itself contain words: it contains the entities indicated by words. Thus meaning, in the sense in which words have meaning, is irrelevant to logic”. (1903a, §51)

This is in many ways an important passage. Firstly, it contains a succinct formulation of Russell’s concept of proposition. Secondly, it contains a similarly compact expression of his conception of meaning: the meaning of a word is an entity which the word stands for. Thirdly, and this is the most important point here, Russell rather 302 Ch. 4 Logic as the Universal Science I bluntly dismisses linguistic meaning as something that is irrelevant to logic. Since propositions are neither linguistic nor psychological entities, meaning in the sense of linguistic meaning is irrelevant to logic. To understand Russell’s philosophy of logic, it is thus necessary to inquire into his notion of proposition. As the quotations from Meinong’s Theory and Principles suggest, Russell’s conception of propositions is derived from Moore. The original source is Moore’s 1898 Fellowship dissertation The Metaphysical Basis of Ethics, whose essential content, as far as the notion of “proposition” is concerned, was published the following year under the title “On the Nature of Judgment” (Moore 1899).

4.4.2 Moore’s Theory of Judgment

The starting-point of Moore’s The Nature of Judgment is Bradley’s notion of meaning. Bradley had argued that empiricist theories of judgment – theories propounded by the likes of Hume, J. S. Mill and Alexander Bain – are vitiated by psychologism: “[i]n England at all events we have lived too long in the psychological attitude. We take it for granted that and as a matter of course that, like sensations and emotions, ideas are phenomena. And considering these phenomena as psychical facts, we have tried (with what success I will not ask) to distinguish between ideas and sensations. But, intent on this, we have as good as forgotten the way in which logic uses ideas”.39 These remarks may be elaborated as follows.40 Any theory that purports to explain our mental life by dint of the mechanism of association of ideas is utterly incapable of explaining the phenomenon of meaningful thought. Insofar as we regard ideas as particular psychical states, we reduce thought to a stream of private occurrences, and facts about these states – facts about the “association of ideas” – cannot explain what judgment, properly understood, is. “The way logic uses ideas”,

39 Bradley (1883, Bk. I, Ch. I, §3). 40 See Gerrard (1997, sec. VI) for a more detailed discussion. Ch. 4 Logic as the Universal Science I 303 according to Bradley, is that here, in accounting for what is really involved when judgment and inference take place, ideas must be taken as symbols, or ideal contents, or abstract universals.41 That is to say, they are entities which are endowed with meanings. When used in a judgment, an idea “must be referred away from itself”,42 i.e., must be taken as an idea of something. This is the bottom-line of the explanation of how a judgment, properly so-called, is capable of truth and falsehood. “Judgment proper”, he explains, “is the act which refers an ideal content […] to a reality beyond the act”.43 In a judgment, according to Bradley, we qualify the real world with a certain adjective – an ideal content – recognizing in the act that the world is so qualified apart from our act.44 The real subject of our judgment is thus reality, and what is correctly ascribed to it in a judgment holds objectively. There is much in these views that Moore could agree with.45 Nev- ertheless, he took Bradley to task for not taking his anti-psychologism far enough. There are reasons to think that he misunderstood Bradley on this point, for he argued that the latter’s account of how meanings or ideal contents are inferentially developed from what is given in our experiences is, in the end, no different from the traditional doctrine of abstraction.46 Be that as it may, Moore concluded that the objects of our thought are in no way a result from any sort of mental activity. And

41 Cf. Bradley (1883, Bk. I, Ch. I, §§4, 6, 7 and 10). 42 Bradley (1883, Bk. I, Ch. I, §3). 43 Bradley (1883, Bk. I, Ch. I, §10). A number of qualifications and corrections to this formulation are made in the 1922-edition of the book. 44 Ibid. The doctrine of judgment, as Bradley explains it in his (1883) is subject to qualifications in the light of his metaphysics. For it turns out that, precisely because the content of a judgment is inevitably something universal (“a wandering adjective”) whereas reality is particular, no judgment is completely true. That, however, is another story; for Bradley’s theory of truth, see Candlish (1989). 45 “[T]o Mr. Bradley’s argument that “the idea in judgment is the universal meaning” I have nothing to add” (Moore 1899, p. 177). 46 On this point, cf. Baldwin (1990, pp. 13-15). 304 Ch. 4 Logic as the Universal Science I this conclusion leads, more or less straightforwardly, to Moore’s notion of judgment:

A proposition is composed not of words, nor yet of thoughts, but of concepts. Concepts are possible objects of thought; but that is no definition of them. It merely states that they may come into relation with a thinker; and in order that they may do anything, they must already be something. It is indifferent to their nature whether anybody thinks them or not. They are incapable of change; and the relation into which they enter with the knowing subject implies no action or reaction. It is a unique relation which can begin or cease with a change in the subject; but the concept is neither the cause nor effect of such change. The occurrence of the relation has, no doubt, causes and effects, but these are to be found only in the subject. (Moore 1899, p. 179)

Moore’s theory is no more than a sketch. However, a number of important points in the theory of judgment emerge from his discussion: (1) “Concepts”, i.e., the constituents of judgments (or propositions; Moore uses the two terms interchangeably), cannot be explained in terms of any “existent fact”. They are the basic metaphysical building blocks from which everything else is composed. (2) The principle of categorial uniformity applies to concepts, that is, all concepts are of the same kind: “the concept turns out to be the only substantive or subject, and no one concept either more or less an adjective than any other” (id., p. 192-193). (3) Propositions are irreducibly relational: “[a] proposition is constituted by any number of concepts, together with a specific relation between them” (id., p. 180).47 (4) Propositions are nothing but complex concepts; they are distinguished from non-propositional concepts by the fact that they are complex, whereas non-propositional concepts are simple (ibid.). This

47 In view of (2), this cannot mean that there is a categorial distinction between concepts and relations. Moore’s position is rather that some among the concepts are relational, others non-relational and all propositions contain at least one relational concept. Ch. 4 Logic as the Universal Science I 305 explains how judgments or propositions can be true or false (ibid.) A simple concept, say the concept red, cannot be true or false for fairly obvious reasons.48 It is only when this concept is conjoined with other concept or concepts that there arises the possibility of truth and falsity: unlike red alone, the combination of red with this house is capable of being either true or false. (5) Complexity is the only respect in which concepts and propositions differ; hence it appears that red house and house is red are one and the same entity that is variously describable as a concept, complex concept or proposition.49 (6) The truth and falsity of propositions cannot be explained, “but must be immediately recognized” (ibid.)50 (7) In accordance with (6), truth and falsity are internal properties of propositions. That is, they are not dependent upon the proposition’s relation to something (“reality”) that is external to it.51 (8) Although the truth and falsity of propositions cannot be further explained, they themselves can be used for metaphysical explanation. For instance, existence is itself a concept, i.e., “something that we mean” (ibid.) Hence, to inquire whether something exists is, meta-

48 Ignoring the usage according to which a predicable is “true” if it applies to something, if there are x’s to which it “truly” applies. 49 This will not do quite as it stands: Red house cannot be a proposition, if – as (3) maintains – all propositions are irreducibly relational. 50 The metaphysics of truth, that is to say, cannot be further explained in the sense that there is no explanation of what truth consists in. Of course, causal and other such common-or-garden explanations can be given as to why this house is red, rather than blue or green, and hence, by Moore’s lights, why the proposition that this house is red is true. Such explanations, however, are not contributions to the metaphysics of propositions. 51 Moore has an argument against truth as correspondence: “[i]t is similarly impossible that truth should depend on a relation to existents or an existent, since the proposition by which it is so defined must itself be true, and the truth of this can be certainly not established, without a vicious circle, by exhibiting its dependence on an existent” (1899, p. 181). This is a familiar argument, Frege being its best-known advocate. I discuss the argument in detail in section 4.5.7.2. 306 Ch. 4 Logic as the Universal Science I physically speaking, to inquire whether a certain proposition is true. This is not to deny that existence is a peculiarly important concept, but it does mean that it is “logically subordinate” to truth (ibid.): “truth cannot be defined by a reference to existence, but existence only by a reference to truth” (ibid.) It is clear that such an amalgam of rather loosely developed ideas cannot be a satisfactory basis for a philosophy of mathematics or philosophy of logic or any other discipline which needs an articulated notion of truth-bearers. This is precisely what Russell needed. Even though Russell’s theory of proposition owes its origin and some of its content to Moore’s theory of judgment, with which it shares a common background ideology (“direct realism”), the propositions of which Russell writes at length in the Principles in fact bear only a rather distant relation to their Moorean ancestors. From Russell’s standpoint, Moore’s notion of proposition is, in fact, multiply flawed. The most significant shortcomings are the following two. Firstly, the notion is restricted to singular propositions, and provides no account of generality, let alone multiple generality. For this purpose, some notion of variable is needed, and Moore offers none. Secondly, Moore’s theory is incapable of dealing with functional dependence; again, the key is provided by the notion of variable. The explanation of these shortcomings is not hard to come by. The fundamental point is that, on Moore’s theory, the composition of propositions is mereological rather than functional. David Bell (1999) has shown in some detail that Moore’s notion of proposition and many of the most characteristic features of his early realism are immediate consequences of a rather bold whole/part theory.52

52 Bell also makes the interesting suggestion that the framework within which Moore develops his early realism is adopted from G. F. Stout’s Ana- lytic Psychology of 1896, which is a detailed presentation of Brentano’s “deskriptive Psychologie”. Moore’s mereology is contained in the following three principles (for details, see Bell 1999, pp. 202-206): 1) the principle of mereological essentialism: if x is a part of W, then W is necessarily such that it has x as part (a whole and its parts are internally related); 2) the principle of Ch. 4 Logic as the Universal Science I 307

At the most general level, Moore’s notion of concept is to be understood in terms of his opposition to idealism, including Kant’s philosophy. According to Hylton (1990a, pp. 172-173), the difference between Moore and Russell, on the one hand, and Kant, on the other, is one between two contrasting notions of objectivity: “[i]s the notion of an object to be taken for granted, as with Moore and Rus- sell, and objectivity explained in terms of it? Or is the notion of an object itself to be explained in terms of objectivity, of which some account is given?” (id., p. 172fn).53 The labels “ontic” and “epistemic” can be used to describe these two competing notions of objectivity. On the epistemic conception of objectivity, judgment is the most natural candidate for that in terms of which objecthood is explained. It is therefore to be expected that one’s stand on the issue of objectivity has implications for one’s views on how the constitution of judgments is to be understood. The Kantian or epistemic conception of objectivity implies a “top-down” approach to the question of constitution. Very briefly, the constituents of judgment, whatever they are, are identified on the basis of the function they perform in judgments. So, in whatever sense of “priority” that is relevant for the explanations of objectivity, mereological adequacy: all forms of complexity involve only whole/part and part/part relations; 3) the principle of mereological atomism: in any complex, the parts are detachable, or each part could exist independently of any whole of which it is a part. For our purposes, the most important of these three principles is the second one, for it rules out irreducibly functional and general phenomena. 53 As Hylton points out (1990a, p. 172fn), the question of objectivity is one that often surfaces in the context of the philosophy of mathematics: many analytic philosophers have had, and continue to have difficulties in taking the notion of an abstract object for granted (the classical discussion is Benacerraf (1973)). The issue is nevertheless a quite general one, as is shown by Michael Dummett’s work on realism vs. anti-realism. A precursor is found in Kant, for whom the problem of the objectivity of mathematical judgments is a special case of a general question about the notion of an object-of-human-cognition. 308 Ch. 4 Logic as the Universal Science I judgments are prior to their constituents. Moore’s object-based or ontic conception of objectivity, by contrast, implies a “bottom-up” approach to the question of constitution. On this view, objects in the most general sense are prior to judgments (or, we could say “propositions”, to emphasize the non-ontic nature of constitution); judgments or propositions are thus to be explained as combinations of these pre-existing elements, which are understood independently of their roles in, or contributions to, propositions. Moore’s theory of judgment reflects in many ways this bottom-up approach. Above all, it shows itself in the characterization of the notion of concept that Moore gives, “concept” being his equivalent to “object in general”. He does not explain concepts by explaining how they function in propositions or judgments (as a Kantian would). In- stead, they receive an abstract characterization as objects of thought and propositions become simply combinations of such concepts (Moore 1899, p. 183). The bottom-up approach is also reflected in Moore’s claim that “the concept turns out to be the only substantive or subject, and no concept either more or less an adjective than any other” (id., pp. 192- 193). This claim is tantamount to a denial that there is a division of entities into distinct ontic kinds or categories; more accurately, it means that there is no valid division of entities into independent (substantival, or substance-like, complete, etc.) and dependent (adjectival, incomplete, unsaturated, etc.) The latter division in particular is most naturally associated with the top-down approach, whose advocates are likely to argue that some such division is needed for a proper understanding of the function of judgments.54 Moore’s attitude here is in keeping with the mereological approach to complexity, which has no use – or, more properly, no

54 The best-known example of this kind of an approach is Frege’s functional approach to propositional composition. Frege makes a great deal of the distinction of propositional constituents into those that are complete and those that are incomplete or “unsaturated”. For further discussion, see below, sections 4.4.6.1 and 4.4.6.2. Ch. 4 Logic as the Universal Science I 309 room – for any such distinction between what is independent and what is dependent. In the context of a theory of judgment or proposition, however, such an attitude can be seen to lead to immediate difficulties, once it is remembered that one of the more important task of a theory of propositions or judgments is to give an account of predication and, relating to that, of truth.

4.4.3 Predication

4.4.3.1 Moore on Predication

There are at least four features that must be incorporated into a theory of predication: (1) the idea of “saying something of something else”, (2) temporary asymmetry; even if one holds that there is no ultimate asymmetry between the elements that figure in an account of predication, each actual instance of predication does introduce a temporary asymmetry between one thing that is predicated and another that is not predicated; (3) the idea that predication gives rise to the possibility of evaluation for truth; (4) unity as the precondition for predication and evaluation for truth.55 The traditional theory of judgment is a straightforward implementation of (1). In judging we are said to unite the predicate with the subject so as to achieve the effect of saying something of something; this is reflected by both the basic logical form of judgment, S – P, and the different modifications of which this form is capable (quantity,

55 I have derived this list, with slight adjustments, from Rauti (2004, pp. 288-289). Rauti presents the traditional idea of “saying one thing of one thing”, the idea of asymmetry and the idea of truth-evaluability as the three most fundamental “intuitions” that pertain to predication or the subject- predicate construction. He does not include unity among these intuitions precisely because he holds (id., p. 289, fn. 29) that it is a product of philosophical observation rather than an articulation of our intuitions. Since no such distinction plays a role in my discussion, unity can be safely included in the list. 310 Ch. 4 Logic as the Universal Science I quality, relation and modality): these are simply different ways of effecting the unification. Moore’s theory is potentially a significant step forward in this respect, though similar steps had already been taken, for instance, by Bolzano (1837) and Frege (1879); in each case the basic idea is that in judging we do not unite or synthesize anything, but recognize something that is already there. It must be said, though, that Moore’s version of this idea was very rudimentary and patently inadequate. This can be seen by considering points (2) - (4).56 The mereological approach to complexity implies that the thesis of categorial uniformity must be accepted in its strongest form, in which it rules out even temporary asymmetry. This idea, however, faces insurmountable difficulties. It is, to begin with, undermined by the intuition that there appear to be complexes that cannot be evaluated for truth. And even if this objection is put aside, the theory must still be regarded as a failure, because it fails to accommodate both (2), the feature of temporary asymmetry, and (4), unity, and therefore also (3), truth.

56 It might be thought that this characterization of the difference between Moore’s theory of judgment and the traditional one jars with what was said above about Moore’s theory implying a bottom-up approach to the constitution of propositions. This, however, is not so. For we must distinguish between the question concerning the constitution of propositions, and the question concerning judgments, which is about the use to which propositions are put. The former question is variously describable as semantic, logical or ontological, depending on how one thinks about such issues. The latter question, by contrast, is epistemic and concerns our cognition. There is no contradiction in holding, as Moore did, that propositions are to be understood on the basis of the independently understood notion of concept, while also holding that, when it comes to the use of these propositions in our cognitive life, they are “prior” to their constituents. Ch. 4 Logic as the Universal Science I 311

4.4.3.2 Russell’s Criticisms of Moore

Russell did not fail to register these difficulties with Moore’s early notion of proposition. In a letter to Moore, dated 1 December 1898 (and hence before the publication of The Nature of Judgment), he wrote:

I have read your dissertation – it appears to me to be on the level of the best philosophy I know. When I see you, I should like to discuss some difficulties which occur in working out your theory of Logic. I believe that propositions are distinguished from mere concepts, not by their complexity only, but by always containing one specific concept, i.e. the copula ‘is’. That is, there must be, between the concepts of a proposition, one special type of relation, not merely some relation. ‘The wise man’ is not a proposition, as Leibnitz says. Moreover, you need the distinction of subject and predicate: in all existential propositions, e.g, existence is a predicate, not subject. ‘Existence is a concept’, is not existential. You will have to say that ‘is’ denotes an unsymmetrical relation. This will allow concepts which only have predicates and never are predicates – i.e. things – and will make everything except the very foundations perfectly orthodox. I cannot conceive an answer to your arguments for the priority of the concept and truth to existence – the few comments of Ward and Bosanquet (especially one about the judgment of similarity) shows a gross misapprehension of your meaning. (Russell 1992b, p. 191)

In this quotation Russell is trying to make the following point. The recourse to mere complexity – two or more concepts conjoined with some relation, as Moore would like to have it – is not enough to guarantee the asymmetry between that of which something is said and that which is said of something, which is necessary for the understanding of predication,. Moore’s theory, Russell suggests, must be complemented by the introduction of a special relation of predication (“copula”), which guarantees temporary asymmetry. With this emendation, traditional subject-predicate propositions could be represented by the following schema:

(1) pred[A, B], 312 Ch. 4 Logic as the Universal Science I where A and B are Moorean concepts conjoined by the relation predication (which is, of course, another concept). The closest natural language analogues of the instances of (1) would be propositions like

(2) Piety is predicable of Peter, or more simply,

(3) Peter has piety, where we intend to make clear by the use of “is predicable of” and “has” instead of the traditional copula “is” that what is predicated is not a mere quality, that is, something that can be described as adjectival or dependent upon the other term of the relation. A theory like this was, for a while, Russell’s own preferred solution to the problem of predication. Although the schema pred[A, B] is unquestionably an improvement on Moore’s unarticulated notion of complexity, it faces its own problems. Russell did not fail to appreciate them. In a paper entitled “The Classification of Relations”, which he read to the Cambridge Moral Sciences Club on 27 January 1899 (Russell 1899b), he raises what is probably the most obvious difficulty:

Finally, I must confess that the above theory raises a very difficult question. When two terms have a relation, is the relation related to each? To answer affirmatively would lead at once to an endless regress; to answer negatively leaves it inexplicable how the relations can in any way belong to the terms. I am entirely unable to solve this difficulty, but I am not convinced that it is insoluble. At any rate, the theory seems equally to affect former theories. When a subject has a predicate, is the predicability of the predicate a new predicate of the subject? This question seems to raise precisely the same difficulty for the opposite theory as the former question raised for mine. To solve this difficulty – if indeed it be soluble – would, I conceive, be the most valuable contribution which a modern philosopher could possibly make to philosophy. (1899b, p. 146) Ch. 4 Logic as the Universal Science I 313

The point of this passage can be formulated in terms of our earlier example. Given a homogeneous pair of terms, [Peter, piety], Russell proposes that a special relation of predication be introduced to restore the temporary asymmetry that is required for predication. This is supposed to lead to the proposition pred[Peter, piety]. But here the question arises as to how the proposition differs from a simple enumeration of constituents, like [Peter, predication, piety]. This is the problem of the unity of propositions. Propositions must be genuine unities – as opposed to mere enumerations of their constituents – for otherwise they could not “say” anything and could not be assessed for truth. But if one holds that all the constituents of a proposition are entities like Peter and piety, one seems to confront precisely this problem of unity. Peter and piety, one is inclined to say, are independent and object-like entities (even if they are called “concepts” as Moore does), and if this generalises so as to cover all propositional constituents, the result is a clash with the requirements that predication imposes on propositions. It may be that the problem of asymmetry is resolved by the introduction of suitable relations. But then an explanation must be given of how the putative constituents of propositions combine with one another so as to yield a genuine unity. Merely to add a relation among the constituents of a proposition – even if it is dubbed the relation of predication – is to invite an infinite regress; without relations, the original terms remain separate and cannot yield a genuine unity, i.e. an entity that can be evaluated for truth. To summarize, Moore’s original theory of propositions faces two kinds of problems. Firstly, it appears to be incapable of dealing with functionality and generality. Secondly, it is incapable of providing an adequate account of the complex problem of predication. The notion of proposition that features in the Principles can be profitably seen as an attempt to resolve these two difficulties. I shall first consider the problem of predication – it is in fact a set of problems rather than a single difficulty – and I shall then turn to the problem of functionality/generality. 314 Ch. 4 Logic as the Universal Science I

4.4.4 Moorean and Peanist Elements in Russell’s Theory of Propositions

The theory of propositions which Russell propounds in the Principles and which underlies his logicism is best described as an uneasy compromise between a Moorean, or bottom-up, and a top-down approach to the constitution of propositions. There is, to begin with, the somewhat inchoate set of doctrines that derives, more or less directly, from Moore. The basic notion here is term, which is Russell’s equivalent of Moore’s notion of concept. Russell’s use of the notion of term represents, up to a point, orthodox Mooreanism. The fact, however, that terms are not just self-subsistent objects but also constituents of propositions forces him to introduce modifications so as to render the theory consistent with requirements that are imposed upon it by the constraints that flow from the predication. Secondly, even though the Moorean theory or its modification is Russell’s official theory of propositions in the Principles, he does consider what is clearly an alternative approach to the constitution of propositions. This alternative approach is derived from Peano, and it makes use of the apparatus of variables and propositional functions. As we shall see below, this theory represents the top-down approach and is therefore scarcely compatible with Moorean orthodoxy.57 On the face of it, a still further complication is caused by the fact that Russell seems to have, in addition to the Moorean theory of propositions and Peano’s logic, yet another approach to the composition of propositions. As we have seen, his official view is that words and linguistic meaning are irrelevant to logic. He is nevertheless willing to maintain that “[t]the study of grammar [...] is capable of throwing far more light on philosophical questions than is commonly sup-

57 That there is, in the Principles, a tension between the Moorean “logic” of propositions and Peano’s symbolic logic has not quite received the attention it deserves. Levine (1998) emphasizes the general differences between the two approaches. Rodríquez-Consuegra (1991, Ch. 4) mentions a number of points where the tension is discernible. Ch. 4 Logic as the Universal Science I 315 posed by philosophers” (§46). Thus he is led to hold that a grammatical difference offers “primâ facie evidence” of a philosophical difference, where by “philosophical difference” is meant a difference that shows up in the analysis of propositions (in “philosophical grammar”, as he calls it; ibid.) Why Russell made this linguistic detour is a question to which secondary literature has not given a satisfactory answer. The most plausible explanation, it seems to me, is that his apparent reverence for grammar is something that he uses to distance himself from traditional logicians, who wanted to torture every judgment into the subject-predicate form, as Bradley put it. Yet the importance of this linguistic detour should not, I think, be exaggerated. Russell, as far as I can see, is never willing to base his argument on grammatical evidence; this applies even in those cases – most notoriously in the connection of the so-called “denoting concepts” – where Russell follows “surface-grammar” more closely than is customary in modern logic.58

4.4.5 The Notion of Term

According to the Russell of the Principles, the most fundamental category of entities is the category of terms. As Russell uses the expression, “term” lacks its usual linguistic connotations. Instead, terms are exactly like Moore’s concepts in that they are, first and foremost, objects of thought (this is argued in §47 of the Principles). Like Moore, however, Russell does not consider this a definition of what it is to be a term. For terms are what they are independently of our cognition, any other view being too much of a concession to psychologism (§427). As we should expect by now, this independence or objectivity of terms is secured by a straightforwardly ontological doctrine, to wit, the distinction between being and existence; absolutely every term has being, even if it should happen not to exist, this being the only way in which the objectivity of thought can be secured:

58 For more on this below, section 4.4.9. 316 Ch. 4 Logic as the Universal Science I

Misled by neglect of being, people have supposed that what does not exist is nothing. Seeing that numbers, relations, and many other objects of thought, do not exist outside the mind, they have supposed that the thoughts in which we think of these entities actually create their own objects. Everyone except a philosopher can see the difference between a post and my idea of a post, but few see the difference between the number 2 and my idea of the number 2. Yet the distinction is as necessary in one case as in the other. The argument that 2 is mental requires that 2 should be essentially an existent. But in that case it would be particular, and it would be impossible for 2 to be in two minds, or in one mind at two times. Thus 2 must be in any case an entity, which will have being even if it is in no mind. [...] In short, all knowledge must be recognition, on pain of being mere delusion; Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians. The number 2 is not purely mental, but is an entity which may be thought of. Whatever can be thought of has being, and its being is a precondition, not a result, of its being thought of. (§427)

Not only does Russell hold that absolutely every term has being; he also holds that anything whatsoever is a term. Thus “term” is the widest word in Russell’s philosophical vocabulary, as he explains in §47: “[w]hatever may be an object of thought, or may occur in any true or false proposition, or can be counted as one, I call a term.”59 In these respects terms are exactly on a par with Moore’s concepts. There are, however, certain important respects in which Russell

59 Russell holds, then, that whatever there is is a term. But what is there? The connection between being and term that he explains in §427 does not as such imply any ontological commitments (admittedly, this passage can be read in a different way: see Hylton (1990a, pp. 211-2)). The further question, what these commitments were, is one that, fortunately, need not be addressed here. This question has received a good deal of attention in recent literature, which is not difficult to understand, as it plays a major role in Rus- sell’s transition to the theory of denoting propounded in On Denoting; see, for example, Cocchiarella (1982), Griffin (1996), Landini (1998, Ch. 3), Makin (2001, Ch. 3), Cartwright (2003), Perkins (2007). Ch. 4 Logic as the Universal Science I 317 modifies Moore’s early views (as he himself points out in §47fn.). Although he recognizes that predication forces certain changes upon Moore’s original doctrines, Russell is nevertheless willing to retain the gist of Moorean orthodoxy, when he denies that the asymmetry of predication implies an ultimate duality between subject and predicate (as in traditional logical grammar), or a thing and its attributes (as in traditional substance-based metaphysics), or this and what (as in Brad- ley’s metaphysics). This is the point of his use of “term”. Russell’s ontology, that is to say, recognizes no genuine distinctions of category: every entity, or anything that has being, belongs to the one and only category that is metaphysically fundamental, the category of terms. But precisely because he sees that each particular case of predication requires asymmetry, he modifies the original notion of term qua object-of-thought so as to render it compatible with facts about predication. This modification is the distinction is between things and concepts, which is introduced in §48 of the Principles. It indicates a difference between two ways in which terms may occur as constituents of propositions. To see the difference, consider a simple predication like “Peter is pious”. Its Russellian equivalent can be represented, as a first approximation, as the proposition /Peter is pious/.60 Intuitively speaking, this proposition appears to be about Peter. Russell follows this intuition, which leads him to say that Peter is the logical subject of the proposition.61 What is characteristic of Peter, logically speaking, is that he is capable of only this kind of occurrence in a proposition. This is Russell’s logical criterion for being a thing. The other class of terms consist of con-

60 I shall adopt the convention introduced by Griffin (1980, p. 119) of using slashes to indicate the mention of a proposition or propositional constituent. Like Griffin, I shall omit slashes when the constituent is an “ordinary entity”; thus I shall say, for instance, that Peter, rather than /Peter/ is the constituent of the Russellian proposition /Peter is pious/. 61 Such alternative phrases as “occupies the position of logical subject” and “occurs as a logical subject” are occasionally useful. For a comprehensive analysis of the notion of logical subject in the early Russell, see Coc- chiarella (1980). 318 Ch. 4 Logic as the Universal Science I cepts. They, too, are capable of occurring as logical subjects in propositions. For example, in the proposition /piety is a virtue/ the subject-position is occupied by /piety/, but it is also capable of a different occurrence, which is shown by the fact that the proposition /Peter is pious/ is not about /piety/. Russell’s terminology, it should be noted, is potentially confusing here. In §47 he writes: “Socrates is a thing, because Socrates can never occur otherwise as a term in a proposition: Socrates is not capable of that curious twofold use which is involved in human and humanity.” Here “term” is used in a sense that is different from the generic sense – term qua object of thought – which was introduced above. This ambiguity, however, is understandable. Russell holds that every term in the generic sense of “term” is capable of occupying the subject-position of a proposition, i.e., occurring as a term in this new sense. Indeed, one could argue that there is a very simple and straightforward connection between the two senses. The generic sense is intended to indicate the ontological role of being an object of thought (an entity that is or has being), and this role is in turn reflected in the logical role of occurring as the subject of a proposition, or of being an entity which the proposition is about. Russell’s characterization of the notion of a thing via the notion of logical subject is relatively unproblematic (I shall return to it below). The same cannot be said of the other member of the dichotomy, to wit, the notion of concept.

4.4.6 The Problem of Unity

4.4.6.1 A Fregean Perspective on Predication

Russell, we have seen, holds that every entity is capable of occurring as a logical subject, that is, that every term in the generic sense is also a term in the logical sense. Thus, even though predication does introduce a temporary asymmetry between the constituents of a proposi- Ch. 4 Logic as the Universal Science I 319 tion, this does not indicate any ultimate asymmetry or difference. He considers at some length an alternative view, according to which the asymmetry of predication is an absolute phenomenon: “It might be thought that a distinction ought to be made between a concept as such and a concept used as a term, between, e.g., such pairs as is and being, human and humanity, one in such a proposition as “this is one” and 1 in “1 is a number” (§49). This alternative, however, is one that Russell rejects. His reasons must be spelled out in detail, for they tell us a great deal about his conception of propositions. The passage continues as follows:

But inextricable difficulties will envelop us if we allow such a view. There is, of course a grammatical difference, and this corresponds to a difference as regards relations. In the first case, the concept in question is used as a concept, that is, it is actually predicated of a term or asserted to relate two or more terms; while in the second case, the concept is itself said to have a predicate or a relation. There is, therefore, no difficulty in accounting for the grammatical difference. But what I wish to urge is, that the difference lies solely in external relations, and not in the intrinsic nature of the terms. For suppose that one as an adjective differed from 1 as a term. In this statement, one as adjective has been made into a term; hence either it has become 1, in which case the supposition is self- contradictory; or there is some other difference between one and 1 in addition to the fact that the first denotes a concept not a term while the second denotes a concept which is a term. But in this latter hypothesis, there must be propositions concerning one as term, and we shall still have to maintain propositions concerning one as adjective as opposed to one as term; yet all such propositions must be false, since a proposition about one as adjective makes one the subject, and is therefore really about one as term. In short, if there were any adjectives which could not be made into substantives without change of meaning, all propositions concerning such adjectives (since they would necessarily turn them into substantives) would be false, and so would the proposition that all such propositions are false, since this itself turns the adjectives into substantives. But this state of things is self-contradictory.

As Peter Hylton (1990a, pp. 175-176) points out, the view Russell is criticising in this passage may be profitably compared with Frege’s 320 Ch. 4 Logic as the Universal Science I distinction between objects (Gegenstände) and concepts (Begriffe), which is an essential part of his functional analysis of predication.62 From Russell’s perspective, Frege’s distinction amounts to one between entities which are capable of occurring in propositions only as subjects and entities which are capable of occurring in propositions only as concepts, or predicatively. Frege’s view is thus that concepts are incapable of occurring as subjects, or that they are essentially predicative. In other words, Frege denies that there are any entities which would be capable of that “curious twofold use” that Russell argues is characteristic of the behaviour of what he calls concepts. A Fregean concept is a meaning (Bedeutung) of a linguistic predicate. Since concepts are construed as functions, linguistic predicates also become assimilated into functions. When the argument-place of a predicate, like “horse ( )”, is filled with a proper name, like “Bucephalus” or “Rosinante”, the result is a complex expression or complex proper name, “horse(Bucephalus)” or “horse(Rosinante)”, which gets evaluated, semantically speaking, along two dimensions. Firstly, it denotes or “means” (bedeutet) a truth-value. Secondly, it expresses the thought which is the condition under which the truth- value is (identical with) the True. The complex expression resulting from the combination of predicate and an object-name is thus a proper name which denotes a truth-value, and it is turned into a sentence by asserting it. Thus “horse (Bucephalus)” denotes the truth- value of: that Bucephalus is a horse, whereas the sentence is an assertion to the effect that this truth-value is (identical with) the True.63

62 Russell himself mentions the similarity in the Appendix A to the Prin- ciples; see §§480-482. 63 If we use a notation that is similar to the one that Frege used in his mature semantics, this paragraph may be rewritten as follows. The meaning of a predicate ‘– ƶƮ’ is a function name which means the True if Ʈ is ƶ, and means the False otherwise; also, the predicate expresses a sense which, when combined with a sense of a proper name, yields a thought denoting the truth-value of: that Ʈ is ƶ. A complex proper name like ‘– f(a)’ is turned into a sentence by attaching a judgment-stroke to it. Thus, ‘»î f(a)’ is an assertion to the effect that the truth-value of: that a is f is the True. This is explained Ch. 4 Logic as the Universal Science I 321

To say that concepts are essentially predicative is to say that this syntactico-semantic pattern holds generally: every subject-predicate sentence, or sentence which “says something of something,” is to be analysed in accordance with the pattern of function and argument. This gives rise to the observation, first made by Benno Kerry, that Frege’s notion of concept excludes the possibility of statements which are about concepts. To take Kerry’s famous example, Frege is committed to holding that the concept horse is not a concept. That is to say, the essential predicativity of concepts means that there are, in Frege’s concept-script, no well-formed sentences about this or that concept. In the sentence “the concept horse is not a concept”, the argument (“the concept horse”) would have to be an expression whose meaning is a concept. However, the expression “the concept horse” is not used predicatively in the above sentence. Therefore, it cannot be a concept-word; it is a complete expression, a proper name, and its meaning is an object.

4.4.6.2 Comparing Frege and Russell on Predication

Two important points can be derived from a comparison between Russell and Frege. Firstly, Frege thinks that his analysis of predication helps resolve the problem of propositional unity. In On Concept and Object Frege considers an alternative analysis of predication, which is essentially the analysis that is found in the Principles:

Somebody may think that this [sc. the paradox of the concept horse] is an artificially created difficulty; that there is no need at all to take account of such an unmanageable thing as what I call a concept: that one might think, like Kerry, regard an object’s falling under a concept as a relation, in which the same thing could occur now as object, now as concept. The words ‘object’ and ‘concept’ would then serve only to indicate the different positions in the relation. (1892, p. 193) in greater detail in §32 of Grundgesetze; see also Frege (1891, pp. 148-156). For a useful discussion of Frege’s mature semantics, see Greimann (2000). 322 Ch. 4 Logic as the Universal Science I

On the proposal under consideration, the distinction between concepts and objects is not absolute (categorial), but pertains only to each individual case of predication. Frege dismisses this Russellian analysis, because he thinks that it renders the problem of unity unsolvable: not all the parts of a thought can be complete, he argues; at least one must be ‘unsaturated’ or predicative, for otherwise the parts of the thought would not “hold together” (ibid.)64 Frege goes on to argue that the asymmetry between the complete and the incomplete or predicative can only be shifted, but it cannot be avoided. He argues that we can dispense with concepts in his sense only if we introduce relations as entities that do the linking. But this move would be of no avail, for the original problem would now arise for relations qua predicative links; since this manoeuvre achieves

64 Frege makes explicit his commitment to the top-down approach to the constitution of propositions, when he explains, in a letter addressed to An- ton Marty or Carl Stumpf that “I do not believe that concept formation can precede judgment because this would presuppose the independent existence of concepts, but I think of a concept as having arisen by decomposition from a judgeable content” (Frege 1980, p. 101). That is to say, the two fundamental categories of concept and object are explained as resulting from the decomposition of complete thoughts. Hylton (1990, p. 173fn) argues that the top-down approach is itself enough to dissolve the problem of propositional unity: “[o]nly those who take objects (in the most general sense) as fundamental are faced with the problem of saying how they combine to form propositions or judgements. This cannot be a question for those who, with Kant and Frege, take the concept of an object to be derivative upon the fundamental notion of a complete proposition or judgement (Fregean Ge- danke).” I do not think that the problem is quite so easily disposed of, even on the top-down approach. For even if one holds that complete propositions are more fundamental than their constituents, one must still explain how the proposition is divided into constituents. At any rate, such an explanation is needed if one thinks that an activity of analysis – decomposition of propositions – is important, as Frege did. Frege’s own struggles with the problem of unity show that he disagreed with Hylton’s appraisal of the conceptual situation. Ch. 4 Logic as the Universal Science I 323 nothing, we might as well return to the position from which we started and invest concepts (“1-place relations”) with predicative power (ibid.)65 It should be emphasised that the proposal which Frege considers is not exactly the position that Russell came to advocate in the Princi- ples. Russell was inclined to think that propositional unity is always due to relations. And he managed to convince himself that the Brad- ley-style objection which threatens with an infinite regress is in fact quite harmless. He admits that a relational proposition /aRb/ implies an infinite array of further relational propositions – /aR´R/, /RR´´R´/, etc. – but he argues (in sections 55 and 99) that, since these further propositions are merely implied and their constituents are not among the constituents of the original relational proposition, the ensuing regress is in fact entirely harmless.66 Since, however, these further relations – R´, R´´, etc. – are merely “implied” and are not among the constituents of the original proposition, they cannot serve as the unifiers of the original proposition (at that time Russell thought that anything that could effect the unity must be among the constituents of the proposition). His considered view, as regards the problem of unity, is that in a relational proposition, /aRb/, the constituent that is responsible for unity is the relation /R/, which occurs predicatively, or as concept, in this proposition. To use Russell’s own phrase, propositional unity is due to “rela-

65 That is to say, Frege considers a proposal that is identical with the proposal that Russell made in his letter to Moore. Given a predicative proposition /Fa/, we deprive the concept of its adhesive power, and since this leaves us with two entities (“terms”) that do not hold together, we introduce a binary predicative relation to reconnect /F/ and a. The result of this operation is the proposition /aƶF/. Obviously, we now treat the relation of predication as the incomplete element, and the original problem reappears. 66 There is of course also the further point that the Russell of the Princi- ples, unlike Bradley, did not find anything per se objectionable in the notion of an infinite regress. 324 Ch. 4 Logic as the Universal Science I tions actually relating” (§49). How is this solution extended to non-relational propositions, that is, propositions of the form /Fa/? This question is difficult to answer, because Russell appears to be incapable of making up his mind on whether there are such propositions in the first place. On the other hand, his Mooreanism inclines him towards the view that there are no non-relational propositions; that even in apparent subject- predicate propositions there really occurs a relation which does the uniting. As was mentioned above, this was at one time Russell’s preferred position, and there certainly are at least vestiges of it in the Principles. The clearest passage in this respect is in §216: “[f]or the so- called properties of a term are, in fact, only other terms to which it stands in some relation; and a common property of two terms is a term to which both stand in the relation”. This is the view that is familiar from Russell’s letter to Moore, and it can be easily made consonant with the points about predication for which Russell argues in the Principles. All we need to do is treat the ordinary constituents of propositions as terms. Predication and unity are then restored by assigning these tasks to special predicative elements. On this suggestion, /Fa/ would be replaced by /ƶ2Ra/, /Rab/ by /ƶ3Rab/, etc., where ƶn is the relation of predication, one for each adicity. Of course, the different ƶns may occur not only as concepts (predicatively), but also as terms or logical subjects, as in the proposition /ƶ2 is a binary relation/. Thus the doctrine that every entity is capable of occurring as a term is still retained. The different relations of predication – the different ƶns – would then be the only relating relations that there are, all other relations being just terms. I do not see that this proposal is inconsistent with the requirements that Russell imposes on predication in the Principles. But he does not seem to advocate this view, either. At least, when he discusses the problem of unity and the notion of a relating relation, all his examples of such relations are “ordinary” relations, and he says quite unequivocally that “verbs” in general are characterized by a twofold grammatical form – verb used as a verb vs. verbal noun – which corresponds Ch. 4 Logic as the Universal Science I 325 to the logical difference between occurrence as a concept vs. occurrence as a term; see, for example, §54, where it is said that the constituents of the proposition /A differs from B/, if it is analysed, seem to be merely A, difference and B.67 On the other hand, Russell is also inclined to think that there are subject-predicate propositions or propositions which are genuinely non-relational. In section 53 he suggests that “the true logical verb in a proposition may be always regarded as asserting a relation”, for example by transforming the proposition /A is/ into /A has Being/, but he continues by remarking that the entire question may be purely verbal. It seems, then, that the question is of relatively little importance to Russell. The important elements of Russell’s Mooreanism are, (1) the general notion of proposition, (2) the doctrine of terms, and in particular the view the every term is capable of occurring as a logical subject in propositions, (3) the division of terms into things and concepts, which is a division between different kinds of occurrences that terms have in propositions. This last feature is introduced to resolve the twin-problem of predication and unity.

4.4.6.3 Problems with Russell’s Account of the Problem of Unity

It has been argued that Russell’s solution to the problem of unity gives rise to a serious problem. This brings us to the second of the two lessons that can be derived from a comparison between Frege’s and Russell’s views on predication. As Griffin (1993, pp. 166-167) points out, the problem with Russellian concepts is precisely that,

67 The Moorean account does lead to a problem, though. Since the binary predication relation, ƶ2, is capable of occurring as a term in predicative propositions, i.e., propositions of the form /... ƶ2 —/, it follows that there are propositions in which one and the same constituent occurs both as a term and as a concept. This would constitute at least an embarrassment for the Moorean account. I am not aware, though, that this point plays any role in Russell’s views on predication. 326 Ch. 4 Logic as the Universal Science I because they are capable of a “curious two-fold use”, they do not unify every proposition in which they occur. Like any other term, concepts can occur as logical subjects in a proposition, and then they do not act as unifiers. For example, /love is a relation/ is a proposition and hence a unity, but its unity is not due to /love/, which occupies the subject-position in this proposition. The ensuing problem can be formulated “phenomenologically” or linguistically. It seems undeniable that we can think and talk about relating relations: they appear to be entirely unproblematic objects of thought and talk. As Griffin points out (id., p. 166), the problem with this is that as soon as we start thinking and talking about them, “we turn them into relations considered as terms”. And this, he argues, leads to a dilemma. On the one hand, Russell thinks he needs the distinction between occurrence-as-a-term and occurrence-as-a- concept, because this promises to resolve the problem of propositional unity; on the other hand, as soon as the distinction is made, we must, in order to state it or think about it, treat relating-relations as relations-as-terms. To use Griffin’s example, if we try to say that R (a relation-as-term) is different from R´ (the corresponding relating- relation), we no longer have R´ as a relating-relation. Merely to state that R and R´ are different turns out to be impossible. In the proposition /R differs from R´/ both R and R´ occur as terms. Hence they are not in fact different, or at any rate they do not differ as relations- as-terms are supposed to differ from relating-relations. The Fregean way to resolve this dilemma would be to blame it on the inadequacy of talk or language. On this view, it is admitted that the difference between R and R´ is there but it is declared to be something that cannot be expressed in words. Such a view – which bears an obvious similarity to some of the more elusive doctrines of Wittgenstein’s Tractatus – is of doubtful coherence and intelligibility; at least, it cannot be stated and treated so lightly as Frege did on this point. The most obvious objection is to ask, What is there to prevent us from speaking about relating-relations or other such “predicative entities”? As Frege himself points out (1892, 186), we often need to Ch. 4 Logic as the Universal Science I 327 say something about concepts in logical theory, and we do this by using what appears to be the ordinary assertoric mode. In the present context, however, such objections are less important.68 The real reason why Frege’s way out is not open for Russell is that, for him, the problem is not really linguistic nor phenomenological at all, but straightforwardly metaphysical.

4.4.6.4 Proposition as Facts

To appreciate the difference between Frege and Russell on this point, we must keep in mind that the constituents of Russellian propositions are worldly entities. This means, among other things, that there is not much of a difference between true propositions in the early Russell’s sense and what other philosophers have called facts. On this view, the fact, assuming it is a fact, that Peter is pious is the true proposition that Peter is pious. More simply still, we can omit any mentioning of facts and say that, on this conception of propositions, Peter’s being pious and the proposition that Peter is pious are one and the same thing. This claim must be taken literally. To take it literally is to recognize the metaphysically constitutive role that the conception assigns to propositions. This role (a special case of it, to be precise) is captured by the following Principle of Truth:

(PT) a is F if, and only if, the proposition /a is F/ is true.

In general, a thing’s having a property or being related to other things is for the relevant propositions to be true. (PT) or its generalization is thus not intended as a mere (necessary) equivalence. The import of the principle is not that, for every putative fact, there is a corresponding proposition; (PT) is meant to capture the metaphysically or onto-

68 According to Dummett (1973, pp. 212-22), Frege found a way out of the paradox of the concept horse with the doctrine that such expressions as ´Ʈ is a concept” and “Ʈ is an object” are pseudo-predicates. 328 Ch. 4 Logic as the Universal Science I logically constitutive role of truth: a is F, because there is a proposition, describable in a certain way, that is true (here “because” is intended to be explanatory in the sense, whatever it is, that is appropriate in metaphysics). We may rewrite one special case of (PT) as follows:

(PT*) Peter is pious if, and only if, the proposition in which Peter occurs as term and which says about Peter that he is pious is true.

It is because Peter is capable of occurring as term or occupying the subject-position in a certain true proposition, namely the proposition which asserts that Peter is pious, that he is pious. That is to say, Pe- ter’s being one way rather than another, i.e., his possessing various properties and standing in various relations to other entities, is metaphysically dependent upon there being a suitable array of true and false propositions of which he is the logical subject. It is because truth is in this way metaphysically constitutive that Frege’s way out is not open for Russell. Unlike Frege, Russell must take with utmost seriousness a situation where it appears that something is not capable of being a logical subject.69 He would commit himself to a blatant contradiction if he argued that there are entities of such-and-such kind (e.g. relating relations), while simultaneously denying that there can be (true) propositions about these entities. To what extent did Russell recognize that the notion of “relating- relation”, which is his solution to the problem of unity, has this consequence? The concept of truth is not given much notice in the Prin- ciples, and what I have called its “metaphysically constitutive role” is not identified as such. Nevertheless, his recognition (indeed, insistence) that it is a logical contradiction to say that there are entities which cannot be made into logical subjects and the concomitant rec-

69 Here I simply assume that Frege’s recourse to the awkwardness of language is an intelligible and coherent position, and hence that Frege’s way out is more than an impasse. Ch. 4 Logic as the Universal Science I 329 ognition that there must be propositions of certain particular kind (§49) comes very close, I think, to an acknowledgement that true propositions have a special role to play in his metaphysics. The conclusion lies at hand, then, that Russell’s Moorean or “bottom-up” approach is in principle incapable of resolving the problem of propositional unity. Russell, it seems, came very close to recognizing this, although his official view was in a way more optimistic: “A proposition, in fact, is essentially a unity, and when analysis has destroyed the unity, no enumeration of constituents will restore the proposition. The verb, when used as a verb, embodies the unity of the proposition, and is thus distinguishable from the verb considered as a term, though I do not know how to give a clear account of the precise nature of the distinction” (1903a, §54). Summing up the discussion so far, Russell’s solution to the problem of predication was to draw a distinction between things and concepts. Although this sounds like a perfectly orthodox idea, the way Russell actually implemented it is in fact highly original. The point is well expressed in the following quotation by Rauti: “Russell does recognize a deep difference between things and concepts but does not locate the difference in the things and the concepts themselves, as Frege does. The difference for Russell only emerges within the propositional context as the inability of things to provide propositions with unity (a collection of things cannot be a proposition). Frege and Russell agree that not everything can be in a predicate position because they both see a connection between occupying that position and ensuring unity, but they disagree as to what can occupy subject position. Frege bars concepts from subject position while Russell does not” (2004, p. 286fn. 20). 330 Ch. 4 Logic as the Universal Science I

4.4.6.5 A Way Out for Russell?

Above the point was developed that Russell’s solution to the problem of predication leads to a severe problem. A closer inspection suggests, however, that the solution may be more resourceful than is commonly thought. There is, according to Russell, no intrinsic difference between a term occurring in a proposition as a logical subject, and a term occurring in a proposition as a concept; the difference is thus a matter of there being different kinds of occurrences of terms in propositions. To put the point another way, the thing/concept distinction is a matter of different kinds of relations holding between the terms or constituents of a proposition; as Russell himself puts it, the difference is only one of external relations (§52). Hence, “being a concept” and “being a logical subject” are relational properties which terms possess in virtue of their positions in a proposition. But if this line of thought can be maintained, then it seems that we can, after all, resist our previous argument. Consider, again, Griffin’s example, the proposition /R differs from R´/, in which /R/ is said to be a relation-as-term and /R´/ is said to be a relating-relation. This counter-example ignores the fact that to characterize an entity as a relation-as-term or relating- relation is to characterize it in terms of relations that it bears to other entities. Thus, R and R´, as they occur in /R differs from R´/ are indeed one and the same term. Hence the proposition is simply false. There is a difference between relating relations and relations considered as terms, but this difference is not one that holds between different terms; it is one that holds between terms-as-occupants-of- positions-in-propositions.70

70 But does it not matter that “there can be no propositions about relating-relations”? The answer seems to be that there are propositions which are about relating-relations. We can say, for example, that /R/ is the relating- relation of /aRb/. That /R/ in this proposition does not occur as a relating- relation does not matter; it is the same entity in both cases, and to require that a relation should be capable of occurring in the subject-position qua Ch. 4 Logic as the Universal Science I 331

On the face of it, this line of thought is quite attractive. However, its consequences for Russell’s theory of predication are not entirely felicitous. Above we characterized the thing/concept distinction as one between different kinds of occurrences of terms in propositions. If, however, “being a concept” were simply a matter of occupying a certain kind of a position in a proposition, one would naturally expect all terms to be capable of such occurrence. This, however, is not the case, according to Russell. For he holds that there are terms which never occur as concepts, but occur only as logical subjects. And this strongly suggests that being a thing and being a concept are intrinsic properties of terms, after all (cf. Rauti 2004, p. 286fn. 21). For Russell, such a conclusion would be problematic at least for two reasons. Firstly, it seems to make the distinction between things and concepts a modal one, since the respective properties determine the kinds of occurrence that a term is capable of. There is, however, officially no room for such modal properties in Russell’s ontology. Had Russell ever formulated this problem to himself, he would probably have denied that the modal characterization adds anything valuable or informative to the original, non-modal characterizations of the properties of being-a-thing and being-a-concept. It should be noted, though, that he does use explicitly modal language, when he explains, in §48, the difference between things (like Socrates) and concepts (like /human/ and /humanity/): “Socrates is a thing, because Socrates can never occur otherwise than as term [sc. logical subject] in a proposition: Socrates is not capable of that curious two-fold use which is involved in human and humanity.” But perhaps this is just a slip of the pen on Russell’s part, for the point of the passage could be easily re- formulated in a modal-free idiom: Socrates is thing, because, in every proposition of which he is a constituent, he occurs as a term or logical subject, whereas /humanity/ is a concept because it occurs in some propositions as a term or logical subject and in others as a concept. relating-relation is arguably just to confuse between a term and a position occupied by that term in a proposition. 332 Ch. 4 Logic as the Universal Science I

The second difficulty is that if we agree that the difference between things and concepts is categorial, we seem to reintroduce just the sort of dichotomy that Moore sought to do away with in his theory of propositions and concepts. It is debatable, however, just how germane the Moorean view is to Russell’s position. In section 49 of the Principles Russell declares that “the theory that there are adjectives or attributes or ideal things, or whatever they may be called, which are in some way less substantial, less self-subsistent, less self-identical, than true substantives, appears to be wholly erroneous, and to be easily reduced to a contradiction”. But the view that Russell is rejecting here is merely that there are things which are not terms in the full- blown sense of being logical subjects; if what has been said above is correct, however, he is merely committed to the weaker view that entities or terms are divided into two kinds in virtue of their “predicative behaviour”: all terms are capable of occurring as logical subjects; only some terms are capable of occurring as concepts (and here “capable of” may be subject to a non-modal paraphrase). Since the asymmetry of predication must be acknowledged in any case, the original Moorean position cannot be maintained in its purity. The third difficulty is by far the most serious one. For there are reasons to think that the modified theory cannot really account for propositional unity. Very briefly, the problem is this: how can the position which an entity occupies in a proposition be responsible for the unity of that proposition? There is a threat of explanatory circle here: we wish to explain what propositional unity is by drawing on the supposedly independently understood distinction between different kinds of positions that terms can occupy in a proposition. More important than this, perhaps, is the doubt, relating to the metaphysics of propositions, that a mere position in a proposition cannot act as a unifier. If we maintain that this position is a matter of a suitable relation or relations holding between the constituents of the proposition (as it seems to be, if we follow Russell’s explanations), then it appears that we are making unity dependent upon relations; this, however, is precisely the sort of explanatory move that Russell tries to avoid, for Ch. 4 Logic as the Universal Science I 333 it would return us to Bradley’s original worries about ‘external relations’ as unifiers.71 Whether Russell has adequate replies to these difficulties is a difficult question which cannot be decided here. I have tried to indicate that there are at least some reasons to think that the prospects of Russell’s theory of predication may not be as dim as is often thought. At any rate, the theory appears to hold out against the most blatant accusations of inconsistency or inadequacy.72

4.4.7 Russell’s Notion of Assertion

Predication and unity are not the only problems which Russell addresses in this theory of propositions. A further set of issues is raised

71 Nicholas Griffin pointed out to me that there is a fourth difficulty. If the constituents of propositions divide into things and concepts and if these can be identified on the basis of the position that an entity has in a proposition, it follows that a proposition cannot be just a combination of terms; propositions must also involve a sequence of positions that the terms can occupy. Furthermore, these positions must be “marked” as restricted in certain ways against particular kinds of term-occurrences, so that things cannot occupy concept-positions. Russell did come to adopt this kind of theory later; the development is clearly visible in the Theory of Knowledge-manuscript (1913), where the constituents of “complexes” – Russell no longer believed in propositions as single entities – have forms, which are characterized, among other things, by the possession of “relative positions” (id., p. 113; Russell discusses the idea in more detail on pp. 145-8). There is probably a great deal to be said about the notion of “position” and its role (either implicit or explicit) in the development of Russell’s views on complexes and unities, but these themes should not be explored here. 72 According to Leonard Linsky, the Russell of the Principles was “defeated” by the problem of unity (1992, p. 250), while Hylton argues that this problem is “in principle unsolvable within the metaphysical framework which he establishes” (1984, p. 381). These judgments may turn out to be correct in the end. I wish to maintain, however, that the issue is rather less clear-cut than is often thought. 334 Ch. 4 Logic as the Universal Science I by the difficult notion of assertion. This is introduced in §38 of the Principles through considerations that are familiar from Frege’s work (Russell, however, arrived at the notion of assertion independently of Frege). According to Frege, neither natural language nor the traditional theory of judgment is well equipped to distinguish between judgment and its content. And yet this distinction must be drawn, because it is vital in logical theory.73 The judgment that Peter is pious is linguistically expressed through the sentence “Peter is pious”. Frege would say about sentences like this that the copula “is” serves a dual role in it: it expresses the content of what is judged – the subsumption of Peter under the concept piety, according to Frege’s mature semantics – and carries assertoric force, since it says (or asserts) that Peter is pious. In Frege’s concept-script, these two roles are clearly separated. In Begriffsschrift, §2, he introduces the complex sign

»î A.

In this sign, “î A” is a complex singular term indicating the content of a possible judgment (beurtheilbarer Inhalt) and “»” is the judgment stroke. Thus, whereas “– pious(Peter)” is used merely to indicate predication – what Frege was later to call “subsumption”, i.e., Peter’s falling under a certain concept – “»î pious(Peter)” signifies a judgment: it asserts the content indicated by “î A”. Frege argues that the distinction between unasserted content and assertion is vital for logical reasons. Consider, for example, modus ponens, or the inference “A; if A then B; therefore B”. Does ‘A’ indicate the same proposition in both of its occurrences? If it does not, then modus ponens is vitiated by ambiguity; if it does, then the first occur-

73 See, for example, Frege’s late essay Die Verneinung (Frege 1918-19b), in which the issue is dealt with at length. The point is present in Frege’s writings throughout his career, however; for Frege’s notion of assertion, see Greimann (2000), to which the following is indebted. Ch. 4 Logic as the Universal Science I 335 rence appears to be redundant, which it is not. Frege solves this problem by writing modus ponens in the following form (I do not reproduce Frege’s notation):

»î A »î A Ⱥ B »î B

According to Frege, the very same content, indicated by “A”, which is asserted in “»î A”, occurs in “»î A Ⱥ B”, but it is not asserted in the latter case. Thus, modus ponens is valid, after all, since validity requires only that identity of content is preserved, and assertion does not affect it.74 In §38 of the Principles, Russell almost recognizes Frege’s point. He writes: “It is plain that, if I may be allowed to use the word assertion in a non-psychological sense, the proposition ‘p implies q’ asserts an implication, though it does not assert p or q.” Thus, in “p; p implies q; therefore q”, the first occurrence of “p” is assertoric, whereas the second is not. So far this reproduces Frege’s point, and one would expect that Russell would use it for the same purpose. However, the very next sentence shows that he may have something altogether different in mind (or simply that he is thoroughly confused): “[t]he p and the q which enter into this proposition [sc. the proposition /p implies q/] are not strictly the same as the p or the q which are separate propositions, at least if they are true” (ibid.) If this claim were correct, it would undermine modus ponens, which is scarcely what is intended.75

74 This point about modus ponens is not the only reason why the distinction is needed. Begriffsschrift, §4, is a brief comparison of Frege’s own theory of judgment with traditional logic. Frege shows that many of the most deeply embedded doctrines of traditional logic are predicated on a failure to distinguish judgments properly so-called from what he calls “judgeable contents”; again, these points are elaborated on in the late essay (Frege 1918- 19b). 75 In spite of the temporary lapse into confusion, Russell manages to make a valid point with the help of the asserted/un-asserted distinction, 336 Ch. 4 Logic as the Universal Science I

What Russell is aiming at with the distinction between asserted and unasserted propositions is precisely the distinction between the separate proposition /p/ and the occurrence of /p/ in /p implies q/. In §52 he describes it as the difference between propositions and propositional concepts, i.e., one between cases like /Caesar died/ and /the death of Caesar/; the former is a proposition – an assertion – whereas the latter is merely a propositional concept – something that is unasserted. And yet, as Frege’s point shows, the two must be the very same entity. It must be said that although the distinction towards which Rus- sell gestures in §52 is a genuine one, his discussion is both confused and confusing. There are two reasons for this. Firstly, he fails to keep separate the distinction between asserted and unasserted propositions, on the one hand, and the problem of unity, on the other. Sec- ondly, he believes, like Frege, that there is an intimate connection between assertion and truth(-evaluation). Thus, Russell is under the impression that one and the same notion (the distinction between asserted and unasserted propositions) is capable of resolving no less than three distinct problems: 1) certain problems concerning inference; 2) the problem of unity; 3) the problem of truth. The first and third of these will be discussed below, in section 4.5.7.6. For now it suffices to indicate source of Russell’s confusion(s). Unity, we have seen Russell arguing, is due to relating-relations (or “verbs used as verbs”, as opposed to verbal nouns);76 at the same when he uses it explain away Lewis Carroll’s puzzle about inference (Carroll 1895); I shall return to this point below, see section 4.5.7.6. The mistake in §38 is corrected in section 52, though the correction is not noted as such; there Russell makes the point using his argument about the impossibility of there being entities that are not capable of occurring as logical subjects. 76 Although “meanings in the sense that words have meaning” are not important for Russell, he discusses, in chapter IV of the Principles, what he calls “philosophical grammar”. He expresses the opinion that the study of grammar is “capable of throwing far more light on philosophical questions than is commonly supposed by philosophers” (§46). Thus, “[a]lthough a Ch. 4 Logic as the Universal Science I 337 time, it is natural to think that this very same entity is what distinguishes assertoric from non-assertoric occurrences of propositions: given a separate, asserted proposition, the assertion, it may seem, is due to the fact that the relation (verb) of the proposition is being used as a relating relation (as a verb). This, at any rate, is what Russell argues in §52, until he recognizes, almost in the middle of the argument, that this view will not do. This dialectically complicated passage runs as follows:

By transforming the verb, as it occurs in a proposition, into a verbal noun, the whole proposition can be turned into a single logical subject, no longer asserted, and no longer containing in itself truth or falsehood. But here too, there seems to be no possibility of maintaining that the logical subject which results is a different entity from the proposition. “Caesar died” and “the death of Caesar” will illustrate this point. If we ask: “What is asserted in the proposition “Caesar died”? the answer must be “the death of Caesar is asserted.” In that case, it would seem, it is the death of Caesar which is true or false; and yet neither truth nor falsity belongs to a mere logical subject. [...] There appears to be an ultimate notion of assertion, given by the verb, which is lost as soon as we substitute a verbal noun, and is lost when the proposition in question is made the subject of some other proposition. This does not depend upon grammatical form: for if I say “Caesar died is a proposition,” I do not assert that Caesar did die, and an element which is present in “Caesar died” has disappeared. Thus the contradiction which was to have been avoided, of an entity which cannot be made a logical subject, appears to have here become inevitable. This difficulty, which seems to be inherent in the very grammatical distinction cannot be uncritically assumed to correspond to a genuine philosophical difference, yet the one is primá facie evidence of the other, and may often be most usefully employed as a source of discovery” (ibid.) Thus Russell is lead to discuss three groups of expressions, proper names, adjectives and verbs, assuming that the differences between these show themselves in differences between three different kinds of propositional constituents (the “meanings” of these expressions). This also explains Russell’s somewhat odd usage. He often speaks of verbs, etc. occurring in propositions (as in the quotation below, when he means the entities that are meanings of verbs qua linguistic items. 338 Ch. 4 Logic as the Universal Science I

nature of truth and falsehood, is one with which I do not know how to deal with satisfactorily.

Three observations may be made about this passage. Together they strengthen our previous conclusion about the prospects of Russell’s Moorean theory of proposition. Firstly, Russell wants to explain truth and falsity by relating them to assertion. He writes that they do not belong to a mere logical subject, like /the death of Caesar/. Clearly, this will not do. Truth and falsehood must belong to unasserted propositions; otherwise complex propositions (/p implies q/, /p or q/, etc.) could not be evaluated for truth. Secondly, he starts his explanation of assertion by arguing that assertion is due to a special constituent of a proposition, namely its “verb”. But then he realizes that this will not do: in the proposition //Caesar died/ is a proposition/, /died/ is without assertoric force. Thus, even though the distinction between asserted and unasserted occurrences of propositions is a valid one, Russell does not succeed in capturing this distinction. Thirdly, this explanation is also incompatible with his official explanation of unity. If assertion is due to “a verb occurring as a verb”, and if this is also the constituent that is responsible for unity, it follows that unasserted propositions (like the antecedent and consequent of an implication or the conjuncts of a conjunction) are not unities. Clearly, this will not do – as Russell himself, in fact notes; see §38 of the Prin- ciples.

4.4.8 Peano’s Logic

4.4.8.1 Why Peano is Superior to Moore

It was mentioned above that the early sections of the Principles present a somewhat uneasy compromise between two fundamentally different approaches to the problem of the constitution of propositions. On the one hand, Russell is attracted to Moorean orthodoxy, which implies a bottom-up approach to the constitution of propositions. Ch. 4 Logic as the Universal Science I 339

On the other hand, there is the technical apparatus of Peano’s symbolic logic and, in particular, the notion of propositional function, which is, on the face of it, incompatible with certain fundamental tenets of the Moorean position. This apparatus recommended itself to Russell, because he saw in it a powerful analytical tool, one that could provide him with that insight into the content of mathematical propositions which he had been lacking until then. Moore’s logic – that is, his conception of propositions – was much less promising in this respect. In the Preface to the Principles Russell mentions a number of doctrines which he says he has learned from Moore, and which he argues are “quite indispensable to any even tolerably satisfactory philosophy of mathematics” (1903a, p. xviii). Yet, the fact remains that these doctrines are very general in nature and offer therefore relatively little, if any, help in the actual construction of a viable “grammar” for the propositions of mathematics.77

77 “I have accepted from him [sc. Moore] the non-existential nature of propositions (except such as happen to assert existence) and their independence of any knowing mind; also the pluralism which regards the world, both of that of existents and that of entities, as composed of an infinite number of mutually independent entities, with relations which are ultimate, and not reducible to adjectives of their terms or of the whole which these compose. Before learning these views from him, I found myself completely unable to construct any philosophy of arithmetic, whereas their acceptance brought about an immediate liberation from a large number of difficulties which I believe to be otherwise insuperable” (1903a, p. xviii). The extent of Peano’s influence on Russell is shown, among other things, by Russell’s often-cited statement that the visit to the International Congress of Philosophy was the single most important event in his philosophical development. Thus, in particular, although the importance of Moore’s influence should not be belit- tled, Russell does not say that the revolt from idealism, which took place in 1898, was the most important event of his philosophical life. To this the following reply might be made. It is true that the revolt from idealism took place gradually and was not an instantaneous event to which a single date could be attached. Since – as Griffin (1991, p. 71) notes – things rarely happened smoothly and gradually in Russell’s autobiography, it could 340 Ch. 4 Logic as the Universal Science I

There are two respects in which Peano’s logic is clearly superior to the Moorean notion of proposition. Firstly, it offers ways of dealing with functional dependence, whereas Moore shows no awareness that this needs to be captured in the first place. Secondly, it offers a way of dealing with quantification or generality.78 Again, Moore’s theory of propositions is absolutely silent on how generality ought to be dealt with; there is no indication that he saw that an explanation is needed in the first place. It is true that Moorean concepts are inherently general entities (which is presumably why they are called “concepts”), but no “technical device” is suggested as to how they can be used in the expression of generality. Russell’s 1898 manuscript, An Analysis of Mathematical Reasoning, which is his most Moorean work, is slightly more advanced in this respect. At the beginning of the manuscript, he offers a classification of the types of judgments that occur in mathematics (1898a, p. 173). One type is “judgments asserting necessary connections of contents”. Of such judgments there are two kinds, the first being judgments in which “the contents may be predicates of the same subject”. Examples are the judgments “human implies mortal” and “triple implies numerical”. This is clearly an acknowledgement that some account of generality must be given. However, “Judgments asserting necessary connections of contents” are scarcely an im- be argued that there is a simple psychological explanation as to why he was inclined to emphasize so much the significance of those few days that he spent in Paris in the late July and early August of 1900, rather than the non- instantaneous dissolution of idealism and adoption of realism which took place in 1898-99. However, when one inspects the writings that Russell produced during his Moorean period – the period between the rejection of idealism and the Paris Congress and the discovery, there, of Peano – one can see that he experienced great difficulties, and made relatively little progress, in exactly those respects in which Peano’s logic offered instant solutions; this can be seen by comparing Russell’s 1898-manuscript The Analysis of Mathematical Reasoning, which is in many respects a Moorean work, with The Logic of Relations (Russell 1901d), which was written in October 1900, and which is the first paper that Russell wrote in Peanese. 78 Both of these problems were mentioned at the end of section 4.2. Ch. 4 Logic as the Universal Science I 341 provement upon, say, the logic of Bradley; it must be admitted, I think, that the beginning of Russell’s theory of quantification was very modest. Peano’s symbolic logic is something completely different in this respect. It is true that Peano’s treatment of quantification is still somewhat rudimentary in comparison with what Frege had already done some ten or fifteen years before.79 The essential elements, however, are all there (see Rodríquez- Consuegra 1991, Ch. 3.1.5). Peano was capable of giving a satisfactory treatment of generality by first drawing a clear distinction between class-membership () and class-inclusion ( ), and then combining the latter with the apparatus of quantified variables (ƶx,y,... x, y, ... Ƹx, y,… where ‘ƶx,y, ...’ and ‘Ƹx,y,...’ are “conditions” on x,y,… and subscripted variables indicate that the condition holds for any entity). Our symbol for the existential quantifier, ‘ ’, occurs only in propositions of the form ‘ a’, where a is a class-symbol: thus ‘ a’ reads ‘the class a is non-empty’, but the symbol is only a shorthand for ‘a - = ’, asserting that a is not identical with the empty class.

4.4.8.2 Predication and Propositional Functions

Peano’s logic is undoubtedly an immense improvement upon Moore’s theory of propositions, when it comes to the treatment of functional dependence and generality. Nevertheless, Russell’s starting-point is the notion of proposition, when he sets out to think about these problems. The reason for this must be that Russell could not model predication upon functionality in the manner of Frege. Frege had argued in his Function and Concept that there is a very close connection between logicians’ concepts and what he calls functions. When a function like x2 = 1 yields the value the True for an argument like -1, this can also be explained by saying that -1 falls under the con-

79 Frege, unsurprisingly, did not fail to mention this in his review of Peano’s work (Frege 1897, pp. 375-8). 342 Ch. 4 Logic as the Universal Science I cept: square root of 1, or that -1 has the property that its square is 1 (1891, pp. 145-146). Frege’s strategy is thus to assimilate concepts and properties into functions. Russell, by contrast, resists this kind of strategy, mainly for the reason that a doctrine which in this respect is like Frege’s involves a commitment to the view that concepts are essentially unsaturated. As Landini (1998, p. 64) notes, Russell’s approach to predication is rather in terms of propositions (and what was said above about the notion of truth – cf. section 4.4.6.2 – supports this): to have a property or to stand in a relation is to occur as a term, that is, as a logical subject, in a suitable proposition. This is why Russell’s approach to functionality draws on the technical device of propositional functions, which presupposes the antecedently given notions of proposition and constituent of a proposition. In the Principles, §338, this is explained as follows:

Accepting as indefinable the notion proposition and the notion constituent of a proposition, we may denote by ƶ(a) a proposition in which a is a constituent. We can then transform a into a variable x, and consider ƶ(x), where ƶ(x) is any proposition differing from ƶ(a), if at all, only by the fact that some other object appears in the place of a; ƶ(x) is what we called a propositional function.80

80 In Russell’s approach, non-propositional functions are therefore always derivative from suitable propositional functions. In the Principles this is explained as follows: “If f(x) is not a propositional function, its value for a given value of x (f(x) being assumed to be one-valued) is the term y satisfying the propositional function y = f(x), i.e. satisfying, for the given value of x, some relational proposition; this relational proposition is involved in the definition of f(x), and some such propositional function is required in the definition of any function which is not propositional” (§482). Russell holds, that is to say, that non-propositional functions – in Principia these are called “descriptive functions”: see Whitehead and Russell (1910, p. 31 and *30) – are less fundamental than propositional functions, since every n-ary function is “derivable” from some n+1-place relation (or propositional function). For example, starting from the proposition /6 is the immediate successor of 5/, which contains a binary relation as one of its constituent, we “derive” from it the binary propositional function “y is the immediate successor of x”, and Ch. 4 Logic as the Universal Science I 343

This characterization can be further explained as follows (see §22 of the Principles). Given a proposition, say /Socrates is a man/, we are to imagine one (or more) of its terms or logical subjects being replaced by any term. This yields a class of propositions – /Socrates is a man/, /Plato is a man/, /the number 2 is a man/, etc. – which differ from the original proposition, if at all, only with respect to the term that is regarded as variable. Although Russell does not put the point quite this way, the idea is evidently that a propositional function is that which the propositions of such a class – we may call it a “variation class” – have in common in respect of a variable term.81 Since, however, there are no “variable terms”, all terms being perfectly definite entities, it is more appropriate to put the point by saying that a propositional function is that which the propositions of such a class have in common in respect of a term or terms regarded as subject to variation. The procedure is intended to be fully general, applicable to propositions of arbitrary complexity. This causes some difficulties for Russell, because he has a somewhat idiosyncratic conception of variables as extra-linguistic entities.82 The main idea, however, is clear enough: the term- or logical subject-position of a proposition marks that term which is subject to variation. Letting x mark the variable term or the position occupied by that term, we get a propositional function, , which expresses the type of all the propositions belonging to the variation-class. As this elaboration indicates, the fundamental notions are not “proposition” and “constituent of a proposition” simpliciter, but, rather, “class of propositions of the same type” and “logical subject” or “occurrence as a logical subject in a proposition”. It appears then that propositional functions are not in any sense reducible to the “logic this in turn can be used to define the non-propositional function whose value for a given argument is the unique entity which satisfies the propositional function for that argument. 81 I have adopted this formulation from Erik Stenius’ explanation of Frege’s notion of function: see Stenius (1976), pp. 265-267. 82 See below, sec 4.4.12. 344 Ch. 4 Logic as the Universal Science I of variation”. For the latter in fact presupposes that propositions can be run together in virtue of their being of the same type or exhibiting the same form, i.e., in virtue of the fact that they can be regarded as instances of one and the same propositional function.83 On the other hand, this should not be taken to mean that the composition of propositions is something that can be understood by dint of propositional functions. This possibility is in fact categorically denied by Rus- sell, who argues, in chapter VII of the Principles, that the unsaturated part of the proposition (what is left of a proposition when the term or terms that are chosen for variation are omitted) is not a genuine term occurring in that proposition as a constituent: “[i]t is to be observed that, according to the theory of propositional functions here advocated, the ƶ in ƶx is not a separate and distinguishable entity: it lives in the propositions of the form ƶx, and cannot survive analysis” (§85). This means, of course, that we must distinguish between the constitution of propositions (the terms that are its constituents) and its decomposition (something that is effected with the help of the notion of a propositional function) Summing up the discussion so far, Russell entertains in the Princi- ples two approaches to the content of propositions. Firstly, there is the idea – somewhat Moorean in spirit, though a great deal more sensitive to details and potential difficulties – of analysis in the sense of laying bare the constituents of propositions. Secondly, there is the idea – which derives from Peano, although, again, the actual implementation is decidedly Russellian – that the content of propositions can be decomposed in accordance with the function/argument pattern. This latter idea, however, is purely technical and is no contribution to any of those problems that the philosophical analysis of propositions may present. On the contrary, the notion of proposition

83 That Russell recognizes this is made evident by §86: “[w]hen a given term occurs as term in a proposition, that term may be replaced by any other while the remaining terms are unchanged. The class of propositions so obtained have what may be called constancy of form, and this constancy of form must be taken as a primitive idea.” Ch. 4 Logic as the Universal Science I 345 and propositional constituent, as these feature in the “logic of variation”, can be used to explain, or at least elucidate, the technical device of propositional functions. Here the leading idea is to think of one or more terms of a proposition as being subject to systematic variation, i.e., capable of being replaced by any other term to yield another proposition.84

4.4.9 The Theory of Denoting

4.4.9.1 Introducing Denoting Concepts

The logic of variation brings us to a crucial component of Russell’s theory of propositions, namely his theory of denoting and denoting concepts. The following four-fold characterization of “proposition” may be gleaned from our previous discussion:

1) Propositions are entities which are assessed for truth; 2) Propositions are complex entities whose constituents can be divided into two kinds; more accurately, the constituents of a proposition are entities occupying one or other of two kinds of positions in that proposition; 3) The logical subject (or subjects) of a proposition is that entity which the proposition is about;

84 My exposition and conclusions are similar to those that James Levine presents in Levine (2002). He argues that the early Russell has two notion of analysis: analysis as the discovery of the “ultimate simple constituents of a given proposition” (id., p. 207), and analysis as functional decomposition. What underlies the former notion is Russell’s principle of acquaintance, that is, the thesis that in every proposition which we can understand, all the constituents must be entities with which we can be “acquainted” in Russell’s technical sense of that word. The latter notion is the notion of a type of a proposition. Since we need not apprehend a proposition as the value of this or that particular propositional function, propositional functions cannot be constituents of Russellian propositions (id., p. 209). 346 Ch. 4 Logic as the Universal Science I

4) The concept of a proposition is that entity which does not occur as a logical subject in that proposition.

Consider the proposition /I met Jones/. This proposition is about me and Jones, and these entities are its logical subjects. By contrast, /met/ occurs in the proposition otherwise than as a logical subject; the proposition is not about that relation. Hence /met/ is the concept of the proposition. Not all propositions, however, are like this:

If I say “I met a man,” the proposition is not about a man: this is a concept which does not walk the streets, but lives in the shadowy limbo of logic-books. What I met was a thing, not a concept, an actual man with a tailor and a bank-account or a public house and a drunken wife. Again, the proposition “any finite number is either odd or even” is plainly true; yet the concept “any finite number” is neither odd nor even. It is only particular numbers that are odd or even; there is not, in addition to these, another entity, any number, which is either odd or even, and if there were, it is plain that it could not be odd and could not be even.” (1903a, §56)

There is a difference between a constituent like Jones and a constituent like /a man/. When Jones occurs in a proposition, the proposition is about him. But when that peculiar concept occurs in a proposition, the proposition is not – as a rule – about that concept. This is shown by the fact that the proposition /I met a man/, if true, is so not because I met a concept but because I met an actual man.85 Nev- ertheless, that man – be it Jones or someone else – does not occur in the proposition /I met a man/ as one of its constituents; what occurs in the proposition is a denoting concept, and the proposition, Russell explains in §56, is about whatever it is that the concept /a man/ denotes. (If one has never heard about denoting concepts, one’s first

85 This formulation – the proposition is true, if it is, because I met an actual man – reverses what according to the Russell of the Principles is the true order of explanation. Since its point is merely elucidatory, the formulation will do as it stands. Ch. 4 Logic as the Universal Science I 347 guess might well be that the proposition /I met a man/ is about that man, whoever he may be, that I met. And that is it is how Russell explains it in §51. It turns out, however, that this is not his settled view on the matter; see section 4.4.9.3.) Denoting concepts are thus the one exception to the general rule which holds in Russell’s early theory of propositions, namely that the entity or entities which a proposition is about are among its constituents. It is for this reason that when denoting is first introduced, it receives a negative characterization:

[a] concept denotes when, if it occurs in a proposition, the proposition is not about the concept, but about a term connected in a certain peculiar way with the concept. (§56)

Clearly, this will not do as a definition of “denoting”, for it contains, in addition to being negative, the phrase “connected in a certain peculiar way”, which requires further explanation. A linguistic criterion would be that denoting concepts are the meanings of, or entities indicated by, denoting phrases, or constructions formed from “all”, “every”, “any”, “a” or “a(n)”, “some” or “the” plus a noun or a noun phrase.86 Denoting concepts and the notion of denoting are of immense importance to Russell theory of logic: “[t]he whole theory of definition, of identity, of classes, of symbolism, and of the variable is wrapped up in the theory of denoting. The notion is a fundamental notion of logic, and, in spite of its difficulties, it is quite essential to

86 Even though such a criterion is readily available (and Russell himself makes use of it in the Principles, sections 57 and 58), it must be emphasized that the phenomenon of denoting with which Russell is concerned is fundamentally non-linguistic: “[t]here is a sense in which we denote, when we point or describe, or employ words as symbols for concepts; this, however, is not the sense that I wish to discuss. But the fact that description is possible – that we are able, by the employment of concepts, to designate a thing which is not a concept – is due to a logical relation between some concepts and some terms, in virtue of which such concepts inherently and logically denote such terms (§56; emphases in the original). 348 Ch. 4 Logic as the Universal Science I be as clear about it as possible” (§56). The theory has several applications. I shall mention three.

4.4.9.2 Why Denoting Concepts are needed

[1] Definition. Russell is most of the time inclined to agree with Frege’s view that definitions are nothing but symbolic abbrevia- tions.87 And yet the formulation of useful definitions is often the hardest part in the process of analysis. This fact, Russell argues, is explained by the theory of denoting:

An object may be present to the mind, without our knowing any concept of which the object is the instance; and the discovery of such a concept is not a mere improvement in notation. The reason why this appears to be the case is that, as soon as the definition is found, it becomes wholly unnecessary to the reasoning to remember the actual object defined, since only concepts are relevant to our deductions. In the moment of discovery, the definition is seen to be true, because the object defined was already in our thoughts; but as part of our reasoning it is not true, but merely symbolic, since what the reasoning requires is not that it should deal with that object, but merely that it should deal with the object denoted by the definition. (1903a, §63)

By making use of the notion of denoting concepts, Russell seeks to reconcile the fact that definitions are, theoretically or in within the structure of a mathematical theory, nothing but symbolic abbrevia- tions, with the epistemic fact that their “discovery” is not just a matter of stipulation. [2] The nature of identity. Identity qua relation is a must-have for the logicist Russell. For instance, the definition of one-one relation is one that makes use of identity as a relation: “A relation is one-one when, if x and x´ have the relation in question to y, then x and x´ are

87 Here we may put aside the Moorean notion of definition as analysis, or the breaking down of what is complex into its (ultimate) constituents. Ch. 4 Logic as the Universal Science I 349 identical; while if x has that relation to y and y´, then y and y´ are identical” (§109). Identity, however, is a puzzling phenomenon, and there are at least two problems standing in the way of recognizing identity as a relation. Firstly, there is the metaphysical problem, long since forgotten, as to whether identity can be regarded as a relation at all. For the early Russell, the problem was real, as can be gathered from §64 of the Principles:

The question whether identity is or is not a relation, and even whether there is such a concept at all, is not easy to answer. For, it may be said, identity cannot be relation, since, where it is truly asserted, we have only one term, whereas two terms are required for a relation. And indeed identity, the objector may urge, cannot be anything at all: two terms plainly are not identical, and one term cannot be, for what is it identical with?88 Whether or not there is a real issue here, Russell’s way out is simply to stipulate, as it were, that identity is a relation, and hence that the nature of relatedness does not require two distinct terms (ibid.) But even after this metaphysical problem is resolved, there remains the familiar puzzle posed by identity: if identity is a relation in which x stands only to itself and nothing else, whatever x may be, how can there ever arise a question as to whether it holds in this or that particular case? Or, to put it another way, how can it be worth while ever to assert identity? Russell replies by drawing on the theory of denot-

88 The objector here is not just an imagined person. As Levine (1998, p. 96) points out, this doctrine was accepted by Moore and, following him, by Russell before his encounter with Peano (see also Levine (2001)). For example, Moore argues in his paper Necessity that there are no propositions whose linguistic expressions takes the form “A is A”, for in that case we do not have two different terms, and therefore we do not have a proposition. That is to say, relatedness requires a genuine multiplicity of terms (Moore 1900, p 295). This view is accepted by Russell, who argued in The Classification of Rela- tions that numerical identity (“self-sameness”) does not fulfil the “formal conditions of relations”, namely a plurality of terms. Thus identity must be regarded as being essentially prior to genuine relations; Russell (1899b, p. 140). 350 Ch. 4 Logic as the Universal Science I ing. If we assert that Edward VII is the King, we assert a proposition in which identity is flanked by an actual term (Edward VII) and a denoting concept (/the King/). Another case is one in which identity is flanked by two denoting concepts, as in /the present Pope is the last survivor of his generation/. It is only when identity is applied to ordinary terms that the assertion is perfectly futile: such assertions, Rus- sell says, are never made outside logic-books.89 Assertions of identity are therefore significant in cases where they are made using denoting concepts. What is asserted then is “pure identity”, i.e., the identity of the denoted term with itself, but the proposition which is used to make the assertion does not contain the identity relation but contains a relation between two denoting concepts or between a denoting concept and a term. [3] It is likely that the primary reason why Russell introduced denoting concepts was that he thought that they would enable him to formulate a workable conception of classes. In particular, he came to think that denoting concepts provided him with an explanation of how we can grasp propositions which are about infinite classes:

With regard to infinite classes, say the class of numbers, it is to be observed that the concept all numbers, though not itself infinitely complex, yet denotes an infinitely complex object. This is the inmost secret of our ability to deal with infinity. An infinitely complex concept, though there may be such, can certainly not be manipulated by the human intelligence; but infinite collections, owing to the notion of denoting, can be manipulated without introducing any concepts of infinite complexity.” (§72)

Classes are extensional entities, since their identity is fixed by the identity of their members. They must nevertheless be given to us by dint of their intensions, except in those rare cases where they happen to be finite. The alternative would be to recognize infinitely complex propositions, and although there may be such propositions, they are beyond our grasp: “[t]hus, our classes must in general be regarded as objects denoted by concepts, and to this the point of view of inten-

89 Russell (1903a, §64). Ch. 4 Logic as the Universal Science I 351 sion is essential. It is owing to this consideration that the theory of denoting is of such great importance” (§66).90 Similar remarks apply to the case of the null-class. Suppose classes were purely extensional entities; suppose, that is that they were given, or defined, by the enumeration of their terms. Clearly, the null-class cannot be given in this way, since there would be nothing to give in the first place.91 The null-class, however, is not nothing, but is, on the contrary, a perfectly definite entity. Again, the apparently problematic class can be given or defined with the help of a suitable denoting concept (although it emphatically cannot be identified with it), namely a denoting concept which denotes nothing. This notion is explained as follows (cf. §73). All denoting concepts are derived from class-concepts. And a is a class concept when, and only when is a propositional function. Thus, the denoting concepts associated with the class- concept a will not denote anything if and only if the propositional

90 In §71, however, Russell seems to endorse a different view on the definition of classes. For there the claim is made that the need for inten- sional definitions of classes is purely psychological. Since our cognitive capacities or essentially finite, we are unable to deal with infinite classes except by dint of intensions and denoting concepts. From the logical point of view, however, “extensional definitions seem to be equally applicable to infinite classes”. I do not know what, if anything, should be made of this. A possible rejoinder in the defence of Russell’s earlier self would be to say that the provision of definitions is essentially a human enterprise. At any rate, this is so if we stick to the view that the only legitimate type of definition is symbolic abbreviation. Thus, an “infinitely complex definition” – such as would be involved in an enumeration of the members of an infinite class – is either an oxymoron, or else “definition” is here given a new, essentially metaphysical sense, which has no application in actual mathematics. Either way, the psychological necessity of employing denoting concepts is a real necessity, and calling it “psychological” does not make it disappear. 91 “From the standpoint of extension”, a class is just a collection of terms, and “what is merely and solely a collection of terms cannot subsist when all the terms are removed” (§73). 352 Ch. 4 Logic as the Universal Science I function is false for all values of x.92 A single pattern emerges from the three cases that we have considered. Denoting concepts are useful – both within and outside mathematics – because they are, to use Russell’s own phrase, “symbolic in their own logical nature” (§51), or because they bring about “aboutness-shifting” (Makin 2000, p. 18). A proposition can be about a term or terms without containing the term/terms among its constituents. And this is useful, because there are cases – including propositions reporting definitions, propositions expressing identity, and propositions which are about infinite classes – where the straightforward connexion between “aboutness” and “being a constituent of a proposition” would bring about insurmountable difficulties.

4.4.9.3 Denoting Concepts and Propositional Functions

As the remarks on classes show, there is a connection between denoting concepts and propositional functions. We have, in fact, already encountered this connection in the notion of a variation class. The variation class, it will be recalled, is the class of all propositions which can be regarded as being of the same type or same form as some chosen proposition. In the simplest case we have a proposition – we could

92 This explanation is not without problems, however. Firstly, propositional functions are now treated as primitive, whereas the official view was that generality is to be analysed with the help of denoting concepts. This means that Russell does not have an explanation of what constituents general propositions have. Secondly, he could not, at that time, persuade himself that propositions like “x is a man implies x is mortal for all values of x” and “all men are mortal” are the same proposition (cf. Principles, §§73 and 77); for the two differ, he argued, in that the former is about any x whatsoever, whereas the latter is an assertion about human beings only. Hence, the problem of non-denoting denoting concepts in fact remains. Thirdly, the proposed analysis of non-denoting denoting concepts cannot be automatically extended to all cases. In particular, definite descriptions remain a problem. These problems were not satisfactorily resolved until in On Denoting. Ch. 4 Logic as the Universal Science I 353 call it the “prototype” – and a term chosen for variation.93 Then the substitution of a term, any term, for the variable term yields a proposition which belongs to the variation class and is thus to be regarded as a proposition of the same type or form as the prototype. If we adopt the notation of propositional functions, we use a variable to mark the position which is subject to variation, i.e., replacement by any term. It appears then that there is a close connection between the device of variables and the notion of “any”. In fact, however, Russell goes further than this, holding that the denoting concept /any term/ is the “true or formal variable” (§88), which is the sort of variable that is relevant for propositions of mathematics. This connection between propositional functions and denoting concepts also enables us to explain the nature of the relation which holds between a denoting concept and an entity when the concept denotes the entity. Consider a proposition like /Socrates is a man/. Regarding Socrates as the variable term yields the propositional function . The position indicated by x can now be occupied by a term, and each such replacement yields a proposition, which is either true or false. Russell holds that a denoting concept like /any man/ denotes those and only those terms for which yields a true proposition (it follows, as we saw above, that if the propositional function is false for every value of the variable, the denoting concept fails to denote anything). Since “any”, “some” etc. are supposed to express distinct denoting concepts, he argues that the differences are due to the fact that they denote different kinds of combinations of terms (different kind of objects, as he calls them):

There is, then, a definite something, different in each of the five case, which must, in a sense, be an object, but is characterized as a set of terms combined in a certain way, which something is denoted by all men, every man, any man, a man or some man; and it is with this very paradoxical object that propositions are concerned in which the corresponding con-

93 I borrow the term “prototype” from Russell (1906b, p. 155). 354 Ch. 4 Logic as the Universal Science I

cept is used as denoting. (§62)94

The details of Russell’s explanation, which is set forth in §§59-62 of the Principles, are quite complex, and they are largely irrelevant to our purposes.95 The theory nevertheless deserves an illustration. Russell’s treatment of “any” and its ontic counterpart, the denoting concept /any a/, serves this purpose well. The treatment is similar to theories which suggest that quantification can be understood in terms of conjunctions and disjunctions. Russell’s proposal, however, is more complicated that this simple identification. He argues, for example, that “every”, “all” and “any”, even though all of them have to do with conjunctions, nevertheless indicate the presence of different kinds of conjunctions. For example, he suggests that /all as/ denotes a “numerical conjunction” of terms that are members of the class a, whereas /any a/ denotes their “variable conjunction”. The latter combination is conjunctive, since “any a” is obviously in some sense concerned with every entity of the kind a. The different as are nevertheless not considered collectively, since “any” has the force of “some one, but it does not matter which”. As Russell puts it, “[t]his form of disjunction denotes a variable term, that is, whichever of the two terms we fix upon, it does not denote this term, and yet it does denote one or other of them” (§59).96 The case of “any” indicates quite clearly that Russell’s theory was not a particularly happy innovation. His official view, stated in §62, is that the differences between denoting concepts are due to differences between denoted objects. And yet it seems clear enough that the difference between, say, “any man” and “every man” is not best cap-

94 The exception to this is the definite article, which presupposes uniqueness. It is of some interest to note that, even though he recognises its importance, Russell treats “the” (or its ontic counterpart, the denoting concept /the .../) almost as an afterthought in the Principles. 95 For these details, see Dau (1986). Geach (1962, pp. 51-80) and Landini (1998, pp. 58-62) compare Russell’s theory with medieval suppositio theories. 96 Russell speaks of two terms here, since he is illustrating the theory by applying it to a simple case where the class a consists of two terms only. Ch. 4 Logic as the Universal Science I 355 tured that way. A consideration of “any” leads Russell to say that /any a/ denotes a variable term, as if there were variable terms, i.e., variable entities, in addition to the perfectly definite ones. Clearly, it would be better, at least in this particular case, to speak not of different kinds of denoted objects but of different denotation relations: variable denotation, whatever it turns out to be, is not an obviously incoherent notion in the way that “ambiguous” (arbitrary, etc.) term or entity would be.97 Russell nevertheless rejects this view, presumably because he could not then hold that the denotation relation is a single indefinable notion of logic.98

4.4.10 Russell’s Analysis of Generality

4.4.10.1 General Remarks

The details of the theory of denoting did not receive a satisfactory treatment in the Principles.99 More important than these details, in the

97 The notion of an incomplete or variable or ambiguous or arbitrary object may be a coherent one, insofar as it is restricted to the singular case (i.e. the denotation of “the a”); cf. Fine (1985) and Santambrogio (1990). Russell, however, would have to have more than this, however; not only arbitrary singular objects, but pluralities of such objects. 98 This suggestion is made by Makin (2000, p. 13 fn3). A different view is adopted by Landini (1998, p. 60), who argues that “the most charitable interpretation is to take Russell’s use of ‘combinations of terms’ as a heuristic device to help in clarifying the kinds of denoting concepts.” According to Dau (1986), Russell’s occasional wavering between these two views – that denotation is in every case the same but the denoted object is different, and the alternative view that the differences are due to different denotation relations – is best explained by the hypothesis that chapter VIII of the Principles, in which the theory is explained, in fact contains two versions of the theory of denoting concepts. 99 Russell returned to the problem shortly after completing the Principles. Between the 1903 and 1905 he produced a series a manuscripts (1903c, 1903d, 1903e, 1903f, 1905d, in which he tried to work out a satisfactory se- 356 Ch. 4 Logic as the Universal Science I present context, is the general motivation behind the theory. Russell introduced it mainly to resolve the philosophical problem of generality (the problem raised by the meaning of “any”, as Russell succinctly put it in a letter to Moore, 15 August 1900 (Russell 1992b, p. 202)). Russell’s treatment of generality is yet another case in which the tension between Peano’s logic and the Moorean analysis propositions is visible. On the one hand, Peano’s logic provided means for the expression of generality (quantification), which, though lacking in technical sophistication, were precisely of the sort that one would expect from a logician. On the other hand, Russell thought – or at any rate hoped – that the Moorean notion of proposition, when combined with the theory of denoting concepts, would constitute a satisfactory philosophy of “any”. In the Principles the view is propounded that Peano’s theory – that is, the notion of propositional function and everything that goes with it – is to be used in the technical development of mathematical logic, but that the philosophical truth about generality, i.e., the analysis of ‘propositional function’, would be delivered by the theory of denoting concepts. Or, to put the point more cautiously, he entertained the hope, while he was working on the Principles, that there should turn out to be a division of labour between the technical development of formal logic (along the lines established by Peano) and philosophical foundations of logic; it is the latter that would, with the help of the notion of proposition, deliver, among other things, the truth about general propositions. Although the philosophical analysis of logic is pursued at some length in the early parts of the Principles, it remains, in many respects, in a rather tentative state; this applies to the “philosophy of generality”, too.

mantics for denoting concepts, semantics that would, at the same time, constitute a foundation for logic and logicism (see Russell (1994) for these manuscripts). These developments culminate, of course, in Russell’s most famous philosophical work, On Denoting (Russell 1905e). Ch. 4 Logic as the Universal Science I 357

4.4.10.2 Formal Implication

Peano’s account of generality is encapsulated in the notion of formal implication. The term is not his, but is the name that Russell gave to Peano’s treatment of (what we would call) universal quantification. The essentials of this account have already been mentioned, but they are worth repeating here. Where p and q are two propositions100 with “real variables” or “variable letters” x...z, Peano writes p . x...z . q, which he reads, “from the proposition p, one deduces q, whatever x...z are”.101 In Peano’s notation, then, universal quantifiers occur only as subscripted to the sign for conditional or implication. It is the resulting complex sign, (“ x...z”) to which Russell gives the name “formal implication”, saying that it is convenient to take this as a single indefinable notion “in the technical study of Symbolic Logic” (§12). To what extent it can be analysed in terms of other, more primitive notions, is a question which belongs to the philosophical foundations of logic, but not to the principles of logic. Two questions should be asked about Russell’s use of formal implication. Firstly, there is the question as to whether it can be given a philosophical analysis. The second question is, why should such a notion should be singled out for special attention? As for the second, more fundamental question, it is not just that Russell, like everyone else, needs some means to express generality, universal quantification going hand in hand with conditionality; the important point is that the idea that receives its formal expression in the notion of formal implication is of fundamental importance to Russell’s way of thinking about logic. It is this idea that will be examined, in a preliminary fashion, in this section; the first question will be taken up in the next section.102

100 Russell was to call p and q “propositional functions”. 101 The French original reads “de p on déduit, quels que soient x...z, la q” (Formulaire de mathématiques, vol. 2, p. iii). 102 For a detailed discussion of the role that formal implication has in 358 Ch. 4 Logic as the Universal Science I

In the Principles, §16, Russell gives the following characterization of logic: “[s]ymbolic logic is essentially concerned with inference in general, and it is distinguished from various special branches of mathematics mainly by its generality”. That “inference” here means deductive inference is shown by the footnote that he adds to §16: “I may as well say at once that I do not distinguish between inference and deduction. What is called induction appears to me to be either disguised deduction or a mere method of making plausible guesses”. Insofar as our interest lies in the subject-matter of Symbolic Logic itself, and not in its application to this or that specific domain, and insofar as we think of this subject-matter in the way suggested by Russell, we are led to say that logic consists in the most general principles of inference; and, since inference is here identified with deduction, we may rephrase this formulation, saying that the subject-matter of logic consists in the most general principles of deduction. It is this way of looking at logic that underlies the introduction of formal implication. The trademark of logic, according to Russell, is its unrestricted generality, and this feature is secured by construing its propositions as formal implications. The connection between formal implication and inference is somewhat less transparent. Russell un- hesitatingly reads formal implication as a kind of proposition.103 One might suspect that, in his case, the connection could be explained by pointing out that the term “principle of deduction” is conveniently ambiguous between a logical law (axiom) and rule of inference; after all, such an explanation would seem to be adequate in Peano’s case, witness his gloss on formal implication as “from the proposition p

Russell’s logic, see section 5.9.1. 103 According to the Principles, §12, formal implications are propositions whose “general type” is “ƶ(x) implies Ƹ(x) for all values of x”; of course, Russell does not mean to restrict formal implications to propositional functions of one variable; the general form of formal implications would rather be “ƶ(x, y, z,... ) implies Ƹ(x, y, z,...), whatever values x, y, z,... may have”, which is the form that “typical propositions of mathematics” have (Principles, §6). Ch. 4 Logic as the Universal Science I 359 one deduces the proposition q, no matter how the variables are determined”. Russell, however, was more sophisticated than Peano on this point, and clearly recognized that an acceptable account of logical inference requires just such a differentiation between laws and rules. This issue, as well as the general question of how formal implication is involved in inference, is best deferred until later, for it is not directly about Russell’s notion of proposition in the way that his analysis is.

4.4.10.3 The Propositions of Logic Again

As we saw in section 4.2.2, Russell has two sorts of considerations backing the view that the propositions of pure logic are of implica- tional form.104 Firstly, they must be universally true. Secondly, this universality must be of a kind that does not recognize any distinctions of type or category. The second requirement amounts to this. Given a formula like

(1) (p . q) p, we automatically think of “p” and “q” as propositional variables.105 Russell, by contrast, construes them as term variables, where “term” has its Russellian meaning. This means that they must be taken to range over all entities there are; absolutely any term – anything that can be counted as one – should be substitutable for the variables occurring in (1). This means that not only propositions like

(2) (it is raining . it is shining) it is raining,

104 As in section 4.2.2, I follow Griffin’s exposition (Griffin 1980, pp. 122-5), with a few modifications; these come mainly from Landini (1998, Ch. 2). 105 Here I ignore the fact that we are not in fact likely to think of “p” and “q” here at all as variables but as schematic letters. 360 Ch. 4 Logic as the Universal Science I but also propositions like

(3) (Augustine . Origen) Augustine are legitimate values of (1).106 Given this, we might follow Landini (1998, pp. 43-44) and replace (1) by (1´):

(1´) (x . y) x 107

Here the use of variables which look like the familiar individual variables is intended to capture the fact that Russellian variables do not discriminate.108 The insistence that propositions like (3) are legitimate instances of (1´) raises two problems for Russell. One of them, discussed in section 4.2.2, was as follows. On the one hand, he wants to say (like eve-

106 When “ ” is read in the manner that Russell read it, to wit, as “implies”, sentences (propositions) do not seem like appropriate entities to flank it: “it is raining implies it is shining” seems to be ill-formed. I will return to this problem below... 107 See here Landini (1998, pp. 43-4). This stipulation is not self- explanatory either, but it is less misleading than the use ‘p’, ‘q’, etc. 108 As Landini (1998, p. 44) points out, Russell’s actual practice in the Principles tends to obscure the true nature of Russellian variables. In his presentation of the “propositional calculus”, Russell uses such letters as “p”, “q”, etc. (cf. §18 of the Principles). A present-day reader, unless he or she is cautious, is practically guaranteed to read these as propositional letters. Never- theless, the beginning of §14 makes it abundantly clear that this is not the intended reading: “[t]he propositional calculus is characterized by the fact that all its propositions have as hypothesis and as consequent the assertion of a material implication. Usually, the hypothesis is of the form “p implies p,” etc., which [...] is equivalent to the assertion that the letters which occur in the consequent are propositions. Thus the consequents consist of propositional functions which are true of all propositions. It is important to observe that, though the letters employed are symbols for variables, and the consequents are true when the variables are given values which are propositions, these values must be genuine propositions” (italics added). Ch. 4 Logic as the Universal Science I 361 ryone else) that all values of (1´) are true, or that each proposition exhibiting this particular form is true. On the other hand, this desideratum is undermined by the recognition of (3) as a legitimate value of (1´); (3) cannot be true, because church fathers are not the sort of entities that can stand in the relation of implication, according to Russell. Russell’s way out of this problem, it will be recalled, was to by conditionalization, which is the standard way of translating from many- to one-sorted logic (Griffin 1980, p. 123). Russell, that is, in- serts a condition to (1´), stating that if the values of the variables are propositions, then a further certain condition holds of them, where the initial condition amounts to a technical definition of the notion of proposition in terms of a property, namely that of standing in the relation of implication, that no other term has. When we add to this the further condition that the variables should be bound, rather than free (apparent rather than real, as Russell would have put it), we end up with the following formal implication:

(4) (x x) x, y ((y y) ((x . y) x))109

In §18 of the Principles, (4) is stated as follows (this is Russell’s axiom (5), called “simplification”):

109 In The Theory of Implication Russell found a way of avoiding conditionalization while retaining the doctrine of the unrestricted variable. He now defined “p q” as equivalent with “p is not true or q is true”. On this understanding of implication, the relation can hold between entities that are not propositions. A proposition like “Augustine Origen” comes out true, on account of the fact that Augustine is not a proposition and is therefore not true. It follows, for example, that a tautology like “p p” can stand on its own, without any further condition being added to it, for it is true, no matter whether p is a proposition or not. Russell does not recommend this approach merely for reasons of simplicity. For he points out that the truth of “(p . q) p” or “p (q p)” should not be made dependent upon q’s being a proposition: if p is true, then, “even if q is not a proposition, it must be added that if p and q were both true, p would be true”; see Russell (1906d, *1.2). 362 Ch. 4 Logic as the Universal Science I

(SIMPL) “if p implies p and q implies q, then pq implies p”

Keeping in mind what has just been said about Russell’s construal of variables, we can see that (SIMPL) is like (4) save for an omission of an explicit mention of quantifiers (“for all values of p and q”)110 and the fact Russell uses words rather than a special symbolism in stating the axiom. He explains (SIMPL) by saying that it “asserts merely that the joint assertion of two propositions implies the assertion of the first of the two”; this is best understood as an informal explanation of the import of the axiom, rather than as a precise translation into ordinary language of the assertion made by (SIMPL). That Russell uses “assertion” in the gloss is probably best explained by the fact that he intends to put his axioms to use in demonstration or inference; for he argues that assertion, as opposed to mere material implication (the relation signified by “ ”), plays an irreducible role in inference.111

110 In a footnote to §14 Russell asks the reader to observe that, in the axioms, the implications denoted by “if” and “then” are formal, while those denoted by “implies” are material (id. fn.*). 111 We should note the following complication involved in Russell’s treatment of formal implications. As was pointed out in footnote 101, propositions do not seem like appropriate entities to flank the relation of implication; the entities which stand in the relation of implication should rather be construed as propositional concepts in Russell’s sense, that is, as entities indicated by nominalizations (“verbal nouns”; see §52 of the Princi- ples). If this is correct, implications should be written as in “7’s being greater than 6 implies 2’s being an even prime”, rather than “7 is greater than 6 implies 2 is an even prime”. In arguing that this is the proper construction for Russell, we need not rely on judgments of grammaticality (the fact that “7 is greater than 6 implies 2 is an even prime” is not a grammatical sentence). A genuinely Russellian argument for the conclusion can be made by referring to the notion of assertion; for the distinction between propositions proper (/seven is greater than 6/, /2 is an even prime/) and propositional concepts (/7’s being greater than 6/, /2’s being an even prime/) goes hand in hand with the distinction between asserted and unasserted propositions. Russell suggests on more than one occasion that assertion is tied up with the “verb” of the proposi- Ch. 4 Logic as the Universal Science I 363

It should be emphasized that, even though the propositions of logic are formal implications, this is no more than a necessary condition of their logicality. For a proposition like /all men are mortal/, if construed after the manner of Peano, is equally a formal implication, though it is clearly not a proposition of logic. Russell adds the condition – one that we shall address in detail in chapter 5 – that the logicality of a formal implication is decided by the sorts of constants that occur in it.

4.4.11 Comparing Russell and Frege on the Constitution of Propositions

Russell’s commitment to a philosophy of logic that is broadly Moorean is nowhere better seen than in the doctrine of terms. Its logical counterpart – indeed, its logical core – is the doctrine, discussed above, that there is but one type of variable, individual or term-variable. This doctrine generates two problems, both of them having to do with Russell’s refusal, at this stage, to recognize distinctions of type. One problem, which he disposes of quite easily, is the one discussed in the previous section, namely that variables must both range over all entities and they must in fact be restricted to entities of a certain kinds. As we saw, Russell used conditionalization to resolve this difficulty. The second difficulty is a more serious one; it is a version of the problem of unity. In earlier sections of this chapter it has already been shown that tion; for example, in the proposition /p implies q/ only the implication is asserted, while p and q are unasserted; from this the conclusion should be drawn that the p and q which occur in /p implies q/ are propositional concepts and not full-blown propositions. Not very much hangs on this, however. As we have already seen, Russell did not succeed in developing a coherent notion of assertion in the Principles; accordingly, there is no clearly articulated distinction between proper propositions and propositional concepts. 364 Ch. 4 Logic as the Universal Science I the ontology of terms, where terms are construed like the “concepts” which Moore had posited in his Nature of Judgment, faces serious difficulties in explaining such phenomena as predication and propositional unity. It is to be expected, then, that precisely analogous problems arise for Russell’s construal of the propositions of logic, when it comes to the articulation of propositional contents, articulation which is schematically represented by means by variables. It is enlightening to compare Russell’s position on this point with Frege’s. We have seen that Frege was explicit about his commitment to the top-down approach to the constitution of propositions: the two categories of entities, objects and concepts, are delineated precisely with a view to explaining the composition of contents that can be judged or asserted. This point can be expressed by saying that, for Frege, the primacy of judgments shows itself in the division of variables into different kinds, reflecting the composition of judgments. By contrast, Russell, insofar as he is under the influence of Moorean ideas, is inclined to think that objects (terms) are prior, in some sense, to judgments (propositions), and, therefore, that there is but one kind of variable.112 This difference between Frege and Russell/Moore is captured by the following figure:

112 “Prior” in the sense that the notion of term is explained and understood independently of the role that terms have as constituents of propositions. Ch. 4 Logic as the Universal Science I 365

Frege: proposition Russell/Moore: term/object p p different kinds term-variables of variables p p objects/concepts ?

Figure 4.2 Frege vs. Russell/Moore on the constitution of propositions

The Russell-Moore approach faces some serious difficulties, insofar it is regarded as delivering a general framework for a philosophy of logic. This is indicated by the question-mark in the figure 4.2; it is not clear, to put the point somewhat bluntly, what we are supposed to do with terms and term-variables. The natural expectation would be that they are put to use in the formation of propositions (contents that can be judged or asserted). In this way, however, there lurks the problem of predication/unity: how can such propositions or contents be built out of mere terms? Frege, by contrast, does not have this problem; or, more correctly, he argues that the difficulty is solved by adopting the top-down approach to propositional composition and the concomitant division of intra-propositional entities into objects (complete entities) and concepts (incomplete or unsaturated entities), whose formal counterparts are the different sorts of variables of Frege’s concept-script. Unlike Moore, who seems not to have been aware of the problem of unity, Russell was rather deeply disturbed by it.113 The requirement that there be but one style of variable – requirement that derives its rationale from the essentially Moorean approach to the constitution

113 This much was shown by the discussion in section 4.4.6. 366 Ch. 4 Logic as the Universal Science I of propositions – poses a difficulty, among other things, for the formal treatment of propositions about relations.114 This is the second problem alluded to above. Propositions that are about relations are important for the logicist Russell, since he holds quite generally that the subject-matter of pure mathematics consists in types of relations (cf. §27 of the Principles). Con- sider, then, a relational proposition like /aRb/. As we have seen, Rus- sell’s position in the Principles is that in this proposition, /R/ occurs as a concept, or as a “verb”, which makes it the entity that is responsible for the unity of the proposition. On the other hand, when he formulates a general principle concerning all relations (or any relation) of some particular type, he must treat relations as variables. Or, more properly, he must treat the position occupied by the relation as subject to variation. And since he holds that there is but one type of variable, the /R/ in /aRb/, understood now as a variable, should be capable of being replaced by any term. However, substituting a non-relation for R in /aRb/ destroys the unity of the original proposition. It seems then that the substitution must be restricted to such values of R as are propositions. And in §28 we find Russell stating the following principle, which belongs to what he calls “the Calculus of Relations”:

(5) if xRy implies x´Ry´, whatever R may be, so long as R is a relation, then x and x´, y and y´ are respectively identical.

R, which is here treated as a variable, is restricted to relations. Russell mentions accordingly that the formal statement of such general principles concerning relations introduces the problem of variables with a restricted field (ibid.) He returns to the topic again in §83, where he makes the suggestion that the strategy of conditionalization can be used here, too. Applied to /aRb/, with /R/ variable, this yields

(6) R is a relation aRb.

114 This difficulty is discussed by Landini (1998, p. 68) and Griffin (1980, pp. 129-131). Ch. 4 Logic as the Universal Science I 367

Keeping in mind that Russellian variables are term-variables, this should be rewritten as

(6´) x is a relation a x b 115

Russell argues in §83 that the x in (6´) can be varied unrestrictedly, i.e., that it can take as values both relations and such terms as are not relations. The idea appears to be that (6´) comes out true for those values of the variable which do not satisfy the condition, and this is so in virtue of the falsity of the antecedent; the strategy is therefore the same that is applied in the calculus of propositions. However, since relations are the entities that are supposed to be responsible for propositional unity (provided they are used as relations or concepts or ‘verbs’), /a x b/ is not a proposition for those values of x that are not relations. Conditionalization is therefore a viable strategy only if the consequent is given some interpretation in those cases where x is not a relation (as Russell points out in a footnote to §83). These values, whatever else they might be, are not propositions; they lack the kind of unity which is characteristic of propositions. The suggested strategy therefore undermines the formal treatment of the propositions of logic that Russell gives in §18. How serious is this problem? At least in the context of the Princi- ples it is very serious, as it shows that there is something seriously amiss in the notions of term and of term-variable, as the “Moorean orthodoxy” would construe them.116 The most natural way out would be to stipulate that only those terms that are relations can be substituted for x in /a x b/; more generally, the following restriction on the variation of terms suggests itself:

115 Cf. Landini (1998), p. 68. 116 The term “Moorean orthodoxy” may be slightly overblown, since its only representative ever, as far as I know, is the Russell of the Principles, who formulated the twin-doctrine of unrestricted generality and term-variables, or variables which are not segregated into different types. 368 Ch. 4 Logic as the Universal Science I

(RV) The substitution of terms for terms within a proposition must always result in a proposition.

However, in the so-called substitutional theory, which was a slightly later development, Russell found a way around most of these problems.117 He even found that he could retain the individual or entity variable as the only sort of variable. In the new theory, terms (genuine entities) are substituted for other terms, and the context in which substitution takes place is usually a proposition, although it need not be. In this way, propositional functions (and classes) can be simulated by considering suitable propositions plus the propositions that emerge, when one (or more) of their constituents are seen as subject to variation. Substitution can be so defined that it becomes a perfectly general notion, and not restricted to propositions (as Russell points out in (1906c, p. 168)). Since, however, the notion that gets “simulated” via the operation of substitution is, precisely, that of prepositional function, rather than function simpliciter, Russell in fact assumes that the distinction between propositions and other entities is primitive, not subject to further analysis.118 This new development (which cannot be explored here in detail) means that “proposition” must be accepted among logical primitives, and this can be seen as Russell’s final repudiation of Moorean orthodoxy. In the Principles this conclusion is reached as well, if only implicitly, for there Russell accepts something like (RV), when he explains propositional functions

117 Serious work on the substitutional theory began in 1905, after the discovery of the idea of an incomplete symbol. Russell’s published accounts of the theory are found in (1906a) and (1906b); see also Russell (1906c). For a detailed discussion, see Landini (1998). 118 Later, after he had rejected the substitutional theory, Russell could still retain the doctrine of the individual variable by deciding that the “entity” (“logical form”, as he would call it) which is responsible for the unity of a proposition is not among the constituents of that proposition; this view contrasts with the idea propounded in the Principles, that the verb of the proposition is both a constituent of the proposition and responsible for its unity. Ch. 4 Logic as the Universal Science I 369 with the help of the notion of variation of terms: considering one (or more) of the terms of a proposition as subject to variation yields the notion of a proposition of some particular type or form, and we have seen that Russell must in fact accept this as a primitive notion.119

4.4.12. Russell’s Account of Variables

These considerations bring us to the first of the two questions which were mentioned at section 4.4.10.2: Can the notion of formal implication be analysed or is it a genuinely primitive notion? From what we already know, we can conclude that the latter option is the right one (at least as far as the Principles go); formal implication is one of those cases where a multitude of propositions is brought together by considering them as propositions of the same type or form; and this notion we have seen to be dependent upon a primitive notion of “being of the same type” or “being of the same form”, which in turn are understood with the help of the notion of propositional function. Russell nevertheless wants to maintain that the theory of denoting concepts throws light on the matter, since denoting concepts feature in the logic of variation, which is used to explain functionality and generality. More precisely, Russell’s hope is that denoting concepts could be put to use in the analysis of the variable, which “is from the formal standpoint, the characteristic notion of Mathematics” (§87). Although formal implications cannot be reduced to anything more basic, he holds that they presuppose the indefinable notion of “any term”, which occurs in the characterization of the notion of variation

119 Recall Russell’s explanation of “propositional function” from §338 of the Principles: “[a]ccepting as indefinable the notion proposition and the notion constituent of a proposition, we may denote by ƶ(a) a proposition in which a is a constituent. We can then transform a into a variable x, and consider ƶ(x), where ƶ(x) is any proposition differing from ƶ(a), if at all, only by the fact that some other object appear in the place of a; ƶ(x) is what we call a propositional function”. 370 Ch. 4 Logic as the Universal Science I class introduced above: given a proposition /ƶa/, a may be replaced by any term, and the result is always a proposition. Applying this idea to a random formal implication, like “x is a man implies, for all values of x, x is mortal,” we get the following approximate translation:

(*) Given the proposition /a is a man implies a is mortal/, every proposition that can be obtained from it by substituting any term for a is true.

As this paraphrase indicates, the defining feature of the variable, as it occurs in mathematics, is its unrestricted range. And in §88 we do find Russell distinguishing between what he calls “the true” or “formal” variable from the restricted variable, suggesting that the true variable (note the definite article!) is what is denoted by /any term/.120 It would be more natural – as well as more in line with what is written elsewhere in the Principles – to identify the variable with the denoting concept /any term/, rather than the denoted object. Indeed, this follows straightaway from Russell’s view that the propositions of pure mathematics contain variables and (logical) constants only (see §§6-9 of the Principles); for if the object denoted by /any term/ were a constituent of propositions of logic, that would turn these propositions into infinitely complex entities, and this will not do for reasons explained above.121

120 “Any term is a concept denoting the true variable; if u be a class not containing all terms, any u denotes a restricted variable. The terms included in the object denoted by the defining concept of a variable are called the values of the variable; thus every value of a variable is a constant” (§88). Later on, in the same passage, he writes: “[b]y making our x always an unrestricted variable, we can speak of the variable, which is conceptually identical in Logic, Arithmetic, Geometry, and all other formal subjects. The terms deal with are always all terms; only the complex concepts that occur distinguish the various branches of Mathematics”. 121 Another reason for resisting the identification is the one discussed in section 4.4.9.3, namely, that it is impossible to make literal sense of “variable terms” or “ambiguous individuals”. Interestingly enough, the conclusion of Ch. 4 Logic as the Universal Science I 371

Russell is quick to note that (*) will not do as a general explanation. The problem is that propositions may contain several distinct variables; what is needed therefore is not an account of the variable, but of variables. This observation elicits the following comment:

x is, in some sense, the object denoted by any term; yet this can hardly be strictly maintained for different variables may occur in a proposition, yet the object denoted by any term, one would suppose, is unique. This, however, elicits a new point in the theory of denoting, namely that any term does not denote, properly speaking, an assemblage of terms, but denotes one term, only not one particular definite term. Thus any term may denote different terms in different places. We may say: any term has some relation to any term; and this is quite different proposition from: any term has some relation to itself. Thus variables have a kind of individuality. This arises [...] from propositional functions. When a propositional function has two variables, it must be regarded as obtained by successive steps. If the propositional function ƶ(x, y) is to be asserted for all values of x and y, we must consider the assertion, for all values of y, of the propositional function ƶ(a, y), where a is a constant. This does not involve y, and may be represented by Ƹ(a). We then vary a and assert Ƹ(x) for all values of x. The process is analogous to double integration; and it is necessary to prove formally that the order in which the variations are made makes no difference to the result. The individuality of variables appears

§88 appears to be that the peculiarity of “any” (or /any/) does not lie in the denoted object but in the manner of denotation. The section is worth quoting at some length: “[i]f ‘any number’ be taken to be a definite object, it is plain that it is not identical with 1 or 2 or 3 or any number that may be mentioned. Yet these are all the numbers there are, so that ‘any number’ cannot be a number at all. The fact is that the concept ‘any number’ does denote one number, but not a particular one. This is just the distinctive point about any, that it denotes a term of a class, but in an impartial distributive manner, with no preference for one term over another. Thus although x is a number, and no one number is x, yet there is no contradiction, so soon as it is recognized that x is not one definite term”. This passage may not be ideally transparent, but the message seems reasonably clear: all terms are perfectly definite entities, and insofar as variables imply variability, this is to be found in the denoting concept – in the manner in which /any a/ denotes – and not in the denoted object. 372 Ch. 4 Logic as the Universal Science I

to be thus explained. A variable is not any term simply, but any term as entering into a propositional function. We may say, if ƶx be a propositional function, that x is the term in any proposition of the class of propositions whose type is ƶx. It thus appears that, as regards propositional functions, the notion of class, of denoting, and of any, are fundamental, being presupposed in the symbolism employed. (§93)

Russell’s account of variables is objectual. That is, variables for him are more than just a notational device. The account that is actually given in the Principles remains seriously incomplete, and Russell himself is not unaware of this, admitting that he has left many details unclear and many questions unanswered.122 Peter Hylton has argued that Russell’s failure on this point is a matter of principle, and not just of a faulty execution. According to Hylton (1990a, pp. 215-217), the only reasonable conclusion that can be drawn from Russell’s discussion is that the very idea that “there are variables in more than a linguistic sense” (id., p. 216) is ill- conceived. As he puts it, “Russell’s constant reversion to the view that there are variables which are non-linguistic entities is the source of considerable confusion in his thought” (id., pp. 216-217). I have two comments to make on this claim. Firstly, the early Russell must develop an objectual account of variables. It is true that there is room in his metaphysics of propositions for entities which could be dubbed “symbolic propositions”. These, however, are not linguistic entities, but propositions which have denoting concepts – entities which are “symbolic in their own logical nature” – as their constituents (see §51 of the Principles). Therefore, since variables are needed, they had better be something more than just linguistic entities, i.e. letters like ‘x’, ‘y’, etc. To this it may be retorted, of course, that, if the idea of “objectual variables” is irreme- diably flawed, merely pointing out that he needs them is not much of

122 In §6 of the Principles we find the following comment: “[t]he notion of the variable is one of the most difficult with which Logic has to deal, and in the present work a satisfactory theory as to its nature, in spite of much discussion, will hardly be found”. See also the end of §93. Ch. 4 Logic as the Universal Science I 373 a defence of Russell. It seems, however, that the situation is less clear- cut than Hylton apparently thinks. This brings us to the second comment. It is not clear, to begin with, that the idea of objectual variables is in fact ill-conceived. It must be admitted that the view that variables qua symbols are genuine names, or that they acquire meaning by standing for objects, must be repudiated.123 It is equally clear, however, that variables do symbolize something, or that they must be given a semantics. For otherwise there could be no assignment of truth-values to quantified sentences. Once this is recognized, it is not clear that we could not hold that variables are whatever the letter ‘x’, ‘y’, etc. symbolize, rather than these letters themselves.124 Furthermore, the apparently non-standard nature of Russell’s views on variables is at least somewhat diminished, as soon as it is pointed out – as we did above – that he should have, in fact, identified the variable with a particular kind of denoting concept, rather than the object denoted. Again, this does not turn variables into linguistic entities, but it does make them entities that are by their very nature symbolic. Being symbolic, their meaning must be explained one way or other. And although Russell begins his account of variables with the view that “the variable” is some sort of an object, this cannot be considered his final view. For as soon as he realizes that what he needs is a theory of variables, rather than just the variable, he admits this much: this can be gleaned from the above quotation from §93 of the Principles. Russell, however, does have an extra difficulty here (and it is most likely this problem that Hylton has in mind in his criticism). “At the level of symbolism”, as Hylton puts it, it is easy enough to distinguish between, say, “ƶx, x” and “ƶx,y”. But Russell, who wishes to construe variables as denoting concepts (or, possibly, as the objects denoted by

123 Variables are, indeed, sometimes referred to as “temporary names”, but this is just a way of talking about the assignment of individuals to variables. 124 See Tichý (1988, Ch. 4) for a coherent notion of an objectual variable. 374 Ch. 4 Logic as the Universal Science I these concepts), must find a way of distinguishing between different kinds of such concepts (or objects); as he puts it, “[a] variable is not any term simply, but any term as entering into a propositional function” (§93). But spelling out this idea is just a matter of finding a viable semantics for variables (hence, for quantified propositions). Pace Hylton, there does not appear to be anything per se in Russell’s position that would make it in principle impossible to find such a semantics; it must be admitted, though, that, in the Principles, Russell did not accomplish very much by way of such objectual account of variables.

4.4.13 Conclusions

The composition of propositions is the big philosophical problem besetting Russell’s theory of logic, and he did not come up with a satisfactory solution to it in the Principles.125 Most of the other concepts that he introduces in Part I of the work are similarly problematic, a fact that Russell himself did not fail to see. Since, however, the difficulties that we have considered above are typically problems for philosophical analysis, they do not undermine Russell’s logicist project and are in that sense something that one can live with. For as long as the problems remain strictly philosophical, Russell can always fall back on the technical apparatus of Peano’s logic, simply disregarding its philosophical foundation. In reality, of course, this attitude was not available to him, as the Paradox which he had discovered could not be classified as philosophical in this sense: it showed to him that there was something seriously wrong with the technical, logical apparatus itself (propositional functions and classes). The Paradox, he

125 And, one might add, his subsequent attempts to come up with a satisfactory solution were equally unsuccessful, witness the long series of theories of proposition that he worked out in the following years. See Griffin (1985) and (1993). The latter paper is particularly useful in that it presents the problem of unity one of the big themes for Russell. For a book-length study of the problem of unity, see Stevens (2005). Ch. 4 Logic as the Universal Science I 375 came to think, affected the “very foundation of reasoning”126 and not only our philosophical understanding of that foundation. As we have seen, the philosophy of logic that is found in the Prin- ciples is characterized by a dual attitude: philosophically, Russell is Moorean (by inspiration rather than by doctrine, for he had to work out all the details on his own), while technically he is an advocate of Peano’s logic. It is important to keep these two approaches in mind, when we turn to consider Russell’s version of the universalist conception of logic. For although his own view seems to have been that the former is just meant to provide the philosophical underpinning for the latter, there are at least prima facie reasons to think that the two represent different philosophies of logic.

4.5. Russell’s Version of the Universalist Conception of logic

4.5.1. Preliminary Remarks

Russell’s universalist conception of logic is, on the face of it, firmly grounded in the notion of formal implication. For it is precisely with the help of this notion that he is able to maintain the following three theses about the propositions of logic:

126 The phrase is from the closing sentence of the Principles. Elsewhere in the Principles Russell argues that the Paradox involves “no peculiar philosophy” but “springs directly from common sense” (§105). Either way, failure to come up with a solution to the Paradox is rather more serious than a failure to resolve the problems of philosophical analysis that affect Russell’s notion of proposition; the former undermines reasoning/common sense, while the latter has at most consequences for whatever philosophical use Russell wishes to put his logic. 376 Ch. 4 Logic as the Universal Science I

(P1) The propositions of logic are truths in their own right, i.e., independently of any sort of interpretation; (P2) the propositions of logic are completely general, i.e., they are about everything; (P3) the propositions of logic stand in a first-order relation to reality.

The universalist conception of logic, as it is usually understood, is taken to have (P1)-(P3) as its cornerstones. Moreover, the view that the propositions of logic are true generalizations about everything (about all “terms”, as Russell would have it) is seen as contrasting with conceptions of logic operating with some explicit notion of form, so that the propositions of logic are construed as generalizations according to form. In the context of Russell’s logic, this contrast yields a distinction between the following two formulations of the propositions of logic:

(1) ƶ (x,y, ...) x, y,... Ƹ(x, y, ...) (2) every proposition of the form ƶ(x, y, ...) Ƹ(x, y, ...) is true

Although much of what Russell has to say about the propositions of logic fits (1), the second formulation is by no means foreign to him. (1) appears to be a straightforward assertion to the effect that a certain complex condition which takes the form of an implication holds for every entity, and Russell often reads it precisely in this way. But this is not always the case. For example, the following quotation from the Principles, §42, conforms to (2), rather than (1):

Thus our formal implication asserts a class of implications, not a single implication at all. We do not, in a word, have one implication containing a variable, but rather a variable implication. We have a class of implications, no one of which contains a variable, and we assert that every member of this class is true. This is the first step towards the analysis of the mathematical notion of the variable. (italics added; cf. also §482)

As formal implications are propositions of the form (1), Russell Ch. 4 Logic as the Universal Science I 377 seems to be saying in this passage that they are to be understood as assertions about form, rather than in the straightforward sense. And, of course, the view expressed in the passage is more than just a passing note; as we have seen, it testifies to the presence, in the Principles, of two levels of analysis: Peano’s logic and Moorean, philosophical analysis of this logic. That is to say, insofar as his focus is on the technical development of logic, Russell accepts formal implication as a primitive notion (as in (1) above), and uses it in the construction of logical calculi; on the other hand, when he considers the question whether formal implication can be given a philosophical analysis, he is attracted to the “logic of variation”. And an analysis which makes use of the idea of variation or substitution accords with (2), rather than (1). This can be seen by considering (*), the gloss on a random implication which was discussed above (see section 4.4.12) and is repeated here:

(*) Given the proposition /a is a man implies a is mortal/, every proposition that can be obtained from it by substituting a term for a is true.

Although there is no explicit mention of form in it, (*) is nonetheless a generalisation over propositions according to their form; the expression “every proposition that can be obtained from [the proposition /...a.../] by substituting any term for a” is the Russellian equivalent of the notion of form, the form being, roughly, the class of propositions which can be developed out of the prototype- proposition by varying one of its constituents. (This characterization of form should have to be generalized, of course, so as to make it applicable to cases where more than one constituent is subject to variation, but we need not concern ourselves with this here.) On the face of it, the observation that there is a distinction to be drawn, within the context of Russell’s logic, between (1) and (2) is of considerable importance for understanding his conception of logic. For an advocate of the van Heijenoort interpretation might well argue 378 Ch. 4 Logic as the Universal Science I that the differences between (1) and (2) are exactly of the sort that serves to distinguish between “logic as language” and “logic as calculus”. To be sure, (2) and (*) fall short of the semantic subtleties characterizing the schematic notion of logical form, which informs the calculus conception of logic. It is nevertheless hard to deny that (2) incorporates, albeit perhaps in a rudimentary form, at least some of those features which the van Heijenoort -interpretation associates with the calculus conception. (2) is, as we might put it, essentially meta-perspectival. For it mentions a particular logical form, rather than asserts a proposition possessing a logical form, and it makes an assertion about this particular logical form by dint of the truth predicate, so that logical propositions become, roughly, generalizations over propositions or propositional forms. Furthermore, once we focus on the pattern exhibited by (*), it begins to look as if Russell even had a notion of interpretation available to himself, even if this notion – interpretation by substitution or replacement – is not the same as the modern one, which operates with the more refined notion of a schematic letter. From the standpoint of the van Heijenoort interpretation, we would have to conclude that the two formulations, (1) and (2), are opposed to one another to the extent that they in fact signal the presence of two different conceptions of logic, conceptions that differ over their respective attitudes towards the possibility of semantic theorizing about logic. Whether we think of interpretation in the modern way, or as taking place through substitution, the point of the exercise is the same in both cases, namely, to determine the behaviour of kinds of propositions with respect to their truth-values. This behaviour is then available as the ultimate standpoint from which the very function of logic is to be explained (logical truth and validity). By contrast, the conception of logic that informs (1) identifies the primary task of logic with the provision of general logical laws and principles underlying correct reasoning,127 principles that in this particular case are to be identified

127 Cf. Goldfarb (2001), p. 28. Ch. 4 Logic as the Universal Science I 379 with propositions of the form (1). What makes this approach anti- semantical is the fact that the generality characterizing (1) simply bypasses serious semantics (model-theoretic argumentation). The fact that there is in Russell’s logic room for both (1) and (2) (and (*)) constitutes a prima facie difficulty for the van Heijenoort interpretation. Although it is undeniable that Russell’s conception of logic was universalist in some suitable sense, we should not simply assume that he understood the conception in a way that excludes meta-perspective, semantic reasoning, or reasoning that presupposes the availability of some notion of interpretation, and suchlike. In other words, it is not clear that Russell considered the opposition between (1) and (2) (or (1) and (*)) in the way suggested by the van Heijenoort interpretation; it is, indeed, possible that there is, in Rus- sell’s view, no opposition here at all, let alone two profoundly different conceptions of logic: this latter possibility is at least suggested by the fact he considered Moorean philosophy of logic and Peano’s symbolic logic as complementing each other. In order to reach a verdict on the issues involved here, we shall have to consider a number of questions about the van Heijenoort interpretation, as it relates to Russell’s views. To anticipate my conclusions, I will argue that the distinction between “logic as calculus” and “logic as language”, as it is conceived of by the advocates of the van Heijenoort interpretation, is, indeed, seriously misleading, when it is applied to Russell. To a considerable extent, our task will consist in the clarification of a number of important concepts: once we come to see what is and what is not involved in such notions as “metaperspective”, “semantic reasoning”, “interpretation”, we shall see that the van Heijenoort interpretation offers a much too simplified picture of the real historical situation. 380 Ch. 4 Logic as the Universal Science I

4.5.2. Russell’s Alleged Anti-Semanticism

4.5.2.1 The Fixed Content Argument and the Argument from Uniqueness

Although I made free use of the term “van Heijenoort interpretation” in the earlier sections of this chapter, it is by no means straightforward to identify the anti-semantical argument, or arguments, that this interpretation assigns to the universalist. What is clear is that the universalist is supposed to accept the following two theses:

(UL-1) The laws of logic have fixed content. (UL-2) There is a unique logic encompassing all forms of deductive reasoning.

The import of (UL-1) is in turn captured by referring to features (P1)- (P3), repeated here:

(P1) The propositions of logic are truths in their own right, i.e. independently of any sort of interpretation; Ch. 4 Logic as the Universal Science I 381

(P2) the propositions of logic are completely general, i.e., they are about everything; (P3) the propositions of logic stand in a first-order relation to reality.

There are, in fact, two distinct lines of thought that have been attributed to the universalist in the secondary literature. Allowing a certain amount of idealisation, we may formulate these as two separate anti- semantical arguments, corresponding to (UL-1) and (UL-2); these may be dubbed the fixed-content argument (FCA) and the uniqueness-argument (UA), respectively. The fixed-content argument could be formulated in the following manner:

(1) The propositions of logic are to be understood in accordance with the thesis of fixed content. ((UL-1), i.e., (P1)- (P3)) (2) The thesis of fixed content excludes external perspective on logic. [from (P1)-(P3)] (3) Therefore, no external standpoint is available from which logic can be surveyed. [from (1) and (2)] (4) But “serious semantics” (model theory) presupposes an external standpoint; it presupposes, in other words, the availability of a meta-perspective. (5) Hence, serious semantics (model theory) is not germane to logic. [from (4) and (5)]

According to (FCA), the anti-semantical conclusion is a simple consequence of the universalist construal of the propositions of logic: the generality of the propositions of logic is substantial rather than schematic, i.e., they stand in a first-order relation to reality; therefore, meta-perspective cannot be germane to logic. Semantics, however, is a discipline that requires meta-perspective. In particular, model- theoretic argumentation is based on the technique of disinterpretation 382 Ch. 4 Logic as the Universal Science I and reinterpretation of non-logical vocabulary, something that the thesis of fixed content is almost calculated to exclude. Hence, serious semantics or model-theoretic argumentation cannot be germane to logic. According to the second line of thought, (UA), what excludes semantics, or renders it impossible, for the universalist, is the thesis of uniqueness, or the claim that there is but one logic for all deductive reasoning. An instance of this argument is formulated by Peter Hylton:

The fact that Russell does not see logic as something on which one can take a meta-theoretical perspective thus constitutes a crucial difference between his conception of logic and the model theoretic conception. Logic, for Russell, is a systematization of reasoning in general, of correct reasoning as such. If we have a correct systematization, it will comprehend all correct principles of reasoning. Given such a conception of logic there can be no external perspective. Any reasoning will, simply in virtue of being reasoning, fall within logic; any proposition that we might wish to advance is subject to the rules of logic. (Hylton 1990a, p. 203)

Here the expression “external perspective” is used, but the passage does not discuss the view that the propositions of logic have fixed content. The argument that Hylton is attributing to Russell is not that once the propositions of logic are construed in accordance with the thesis of fixed content, meta-perspective is seen not to be germane to logic. His point is rather that universality in the sense of uniqueness implies the denial of an external perspective or standpoint that falls outside logic. According to this argument, then, there is more to “external perspective” than the relatively innocuous sense of meta- perspective that features in (FCA); in the innocuous sense, external perspective/meta-perspective is simply constituted by statements about a discipline as opposed to statements within a discipline, and these are excluded by the thesis of fixed content. In the argument now under consideration, by contrast, an external perspective on a discipline, X, is constituted by statements that are not only about X but are not themselves part of X, point about which (FCA) appears to Ch. 4 Logic as the Universal Science I 383 be silent. Applied to logic, the possibility of adopting a meta- perspective implies the possibility of stepping outside logic in some as yet unspecified sense; and since this is not possible, according to the universalist, meta-perspective on logic is illegitimate. The version of (UA) that Hylton ascribes to Russell can be formulated as follows:

(1) Logic comprises all principles of correct reasoning. (2) Hence, there is no reasoning outside the one universal logic. (3) Meta-theoretical reasoning presupposes principles which do not belong to the system under scrutiny. (4) Hence, there can be no meta-theoretical reasoning.

Applied to our present discussion, the above argument is incomplete as it stands, as there is no explicit reference in it to semantics. The argument is easily modified, however; what is missing from it is simply the premise that serious semantics is grounded in reasonings that are meta-theoretical in character, and hence that serious semantics presupposes the possibility of stepping outside the system that is being discussed. This yields the following argument, which I shall call (UA-1):

(1) Logic comprises all principles of correct reasoning. (2) Hence, there can be no reasoning outside the one universal logic. (3) Semantic argumentation presupposes principles which do not belong to the system under consideration, or principles that fall outside it. (4) Hence, there can be no semantic reasoning about the universal logic.

In what follows, I shall first consider (UA-1); I will then return to the argument from fixed content. 384 Ch. 4 Logic as the Universal Science I

4.5.2.2 What is Wrong with the Uniqueness Argument

(UA-1) cannot be regarded as a satisfactory reconstruction of the uniqueness-argument. According to the van Heijenoort interpretation, a believer in the universality of logic and an advocate of the calculus conception are supposed to adopt opposing views on the possibility of meta-perspective. However, no genuine opposition is forthcoming, insofar as the reasoning we attribute to the universalist complies with (UA-1). If there is to be a real contrast between the universalist conception and the calculus conception on this point, one of them should presumably accept what the other denies or finds impossible. An advocate of the calculus conception, however, is not likely to argue that the possibility of a meta-perspective on logic presupposes that we be able to station ourselves outside logic, or outside the principles of correct reasoning. What is characteristic of the calculus conception is rather the view that there is no such thing as logic simpliciter; logic is always tied up with this or that particular language for which a deductive system or semantics, or both, has been defined; a natural ally of this view – of which the model-theoretic conception of logic provides an example – is the distinction between a logic and its metatheory and the concomitant distinction between object-language and metalanguage. In other words, an advocate of the calculus conception holds the following:

(Meta) Meta-perspective is possible because it is possible to step outside a particular language or a particular calculus for logic, whereas a believer in the universality of logic is supposed to assert:

(Non-meta) Meta-perspective is impossible because it is impossible to step outside logic. Ch. 4 Logic as the Universal Science I 385

Clearly, there is no contrast between (Meta) and (Non-meta); (Meta) is a thesis to the effect that a language or calculus for logic can be studied from outside, whereas (Non-meta) is about something altogether different, to wit, about the (im)possibility of adopting a standpoint that falls outside the sphere of logic. If there is to be a genuine contrast between the two conceptions, the only reasonable way to articulate it is with the help of (Meta); the questions raised by (Non-meta) are of a different kind altogether, and not at all at issue for an advocate of (Meta). Once we focus on (Meta), however, we can see that it is not ruled out by universality qua uniqueness, or the thesis that logic comprises all principles of correct reasoning. Of course, the early Russell would not have cared about a language for logic, for he did not believe that logic is a matter of this or that language; but as in the above formulation, the thesis can be phrased by speaking about “calculus”, a term which is appropriately neutral with respect to the linguistic turn. Un- derstood in this way, (Meta) is entirely compatible with the claim that there is, in the last instance, only one logic. Gregory Landini (1998, p. 34) gives a succinct formulation of the essential point. In the case of such logical realists as Russell and Frege, we must distinguish between logic as the universal science and logic as a deductive calculus. That is to say, we must distinguish between the science of logic and a particular formulation – a formal system – which is intended to capture that science.128 By way of comparison, Landini (ibid.) points out that no mathematical realist would confuse arithmetic qua science with an axiom system of formal number theory; the same applies to a logical realist, who would insist on drawing an analogous distinction for logic. Consider in this light the above quotation from Hylton. The crucial part runs as follows: “if we have a correct systematization [of logic], it

128 Or less than that science; there is no reason why a universalist could not work out a deductive system for some limited purpose, i.e., without considering the question as to whether the system might capture all the principles of correct reasoning. 386 Ch. 4 Logic as the Universal Science I will comprehend all correct principles of logic. Given such a conception of logic there can be no external perspective” (italics added). The conclusion turns out to be a non sequitur. The important expressions in the passage are the italicized ones: “if” and “correct systematization”. Logic qua universal science is not to be confused with a calculus for logic. Nevertheless, it is also the case that the universal science of logic is not something that is given us; on the contrary, a “correct systematization” of logic is something that must be discovered and formulated (in the form of an axiomatic system, according to Rus- sell), in exactly the same way that a set of axioms must be discovered for arithmetic.129 And here at least a certain amount of meta-theoretic reasoning is not only possible but in fact necessary. A believer in the universality of logic can readily admit that it is impossible to step outside the universal science of logic (outside all correct principles of logic).130 The universality of logic in this sense, the idea that there is but one logic which encompasses all principles of deductive reasoning, does impose certain restrictions on what one can and cannot do in the metatheory for this or that calculus of logic. It does not follow from this, however, that the universalist could not adopt an external perspective on a particular formalism or calculus for logic. Thus, once we make clear to ourselves what is and what is not involved in the possibility of meta-perspective, we can see that there is, indeed, room for an external perspective and meta-theoretic reasoning in the universalist’s scheme of things.

129 Furthermore, a universalist logician is not committed to there being a “correct systematization of logic”; he is at liberty to hold (as both Frege and Russell in fact did) that several axiomatizations are permissible. 130 What, exactly, this impossibility amounts to is a question that can be answered in more than one way. Here I simply assume that the early Russell would have accepted this “impossibility result” in one form or another; I am not aware of any passage in which he engages in an explicit commentary on this somewhat Wittgensteinian, or Fregean, theme. Ch. 4 Logic as the Universal Science I 387

4.5.2.2.1 Did Russell have a “Calculus for Logic”?

As far as Russell is concerned, this criticism of (UA-1) could be re- sisted in two ways. Firstly, one could argue that even if the universalist position, abstractly considered, does allow a distinction to be drawn between logic qua science and a formal calculus for logic, this is not in fact how Russell saw the matter; in other words, the claim is that Russell lacked the idea of a formal calculus for logic. Secondly, one may admit that the universalist conception of logic does admit such a distinction and that it applies to Russell’s case, too, and nevertheless maintain that this conception imposes non-trivial constraints on the meta-perspective that is available to a universalist. In what follows, I shall consider these rejoinders in turn. The first rejoinder, or the claim that Russell lacked a notion of a formal calculus for logic, is maintained by Hylton, who argues that there is in this respect an important difference between Frege’s and Russell’s respective conceptions of logic:

There is [...] one general difference between Russell’s conception of logic and Frege’s. Russell’s conception of logic is based on a metaphysical view which could, and to some extent was, articulated quite independently of logic. Russell [...] held himself to be indebted to Moore for the metaphysics of propositions and their constituents, of being and truth. This metaphysics is independent of the logic which Russell erected upon it [...]. For Russell, then, the metaphysics was independent of and prior to the logic. For Frege, [...] the opposite is true. For Frege, logic, in the sense of the inferences that we do in fact acknowledge as correct, is primary; metaphysics is secondary, and articulated in terms which presuppose logic. (Hylton 1990b, p. 215)

According to Hylton, this very general difference has consequences for Frege’s and Russell’s respective ways of doing logic. For Frege, who believed that logic is independent of metaphysical considerations, the only way to delimit the body of correct reasonings is to do it independently of concepts that belong to metaphysics. More generally, he held the view that since the foundations of logic are inde- 388 Ch. 4 Logic as the Universal Science I pendent of all extra-logical disciplines, all such disciplines – metaphysics, psychology, etc. – are irrelevant to logic.131 It is for this reason, according to Hylton, that “Frege gives something very like a modern syntactic account of logic” (ibid.), and since “syntactic account” is here modified by “modern”, the implication is plainly that Frege operated with a relatively strict notion of formalism. Hylton does not elaborate on this, but I take it that his meaning is approxi- mately as follows. Admittedly, Frege does not formulate his logical system in the way that later logicians would introduce theirs. Never- theless, the logicist program presupposes a notion of rigour, or rigorous reasoning, and the only notion of rigorous reasoning that is available to him is syntactic (for reasons given above). Because of this, Frege’s notion of formal system, even if he does not introduce his concept-script in the modern way, is nevertheless essentially identical with the modern one. Hylton goes on to argue that Russell’s stance is importantly different from Frege’s. The reason for this, according to Hylton, is that the two disagree on how logic and metaphysics are related to each other:

Russell’s philosophical-logical views do not need to be based on a neutral, and therefore syntactic, notion of correct logical inference, for Rus- sell’s metaphysics is independent of logic and therefore available for use in defining the notion of logic. The definition is given in terms of the notion of a proposition, of the constituents of a proposition, and of truth. These notion are [...] ones to which we have direct and immediate access, through a non-sensuous analogue of perception. Thus there is no need for a syntactic approach, from Russell’s point of view. This is not to say that anything in Russell’s conception of logic in fact rules out such an approach, though this conception of logic does show something about the significance of the results which can be obtained in this way. What it does indicate is that the syntactic approach is not a natural one for

131 Frege has of course, available to himself such notions as judgment (as in Begriffsschrift), or thought (‘Gedanke’, as in his mature theory of logic, beginning in the early 1890s); according to the interpretation of Frege that Hylton favours, however, they receive whatever content they have through logical, rather than metaphysical (let alone psychological) considerations. Ch. 4 Logic as the Universal Science I 389

someone with Russell’s conception of logic; there is no particular reason why it should have occurred to Russell. Nor indeed do I think that it did. (Hylton 1990b, p. 216)

Hylton’s complex claim about the differences between Frege’s and Russell’s conceptions of logic and their implications is significant in the present context, for the conclusion of his argument in fact amounts to the claim that Russell lacked a notion of formal system/calculus; or at least that he, unlike Frege, did not have a particular reason to come up with such an idea. If correct, this is obviously relevant to the above criticism of (UA-1). The criticism was essentially that there is nothing in the universalist conception to preclude meta-perspective, once this is understood in terms of the distinction between logic simpliciter and logic qua calculus. As against this, the point could now be advanced that even if the universalist could draw such a distinction, there is no compelling reasons why Russell would have done so. Firstly, he was not like the model-theoretic logician who thinks of logic in terms of formalism plus interpretations. Secondly, and this is the present point, he lacked the sort of reason that Frege had. Hence, there is no particular reason why the distinction between logic simpliciter and logic qua calculus would have occurred to Russell, or would be applicable in his case. Hylton’s reasoning is not convincing. He is correct to argue that Frege’s and Russell’s respective approaches to the foundations of logic were not identical. It seems to me, however, that he overstates the implications of these differences for Russell. He is correct in arguing that Russell’s notion of proposition, which is of central importance to his theory of logic, is not a purely logical one in the way that Frege’s “judgment” or “thought” arguably is. As we have already seen in examining Russell’s “Mooreanism”, the notion of term with the help of which he purports to cash out the notion of proposition, signifies for him a metaphysical rather than logical category; Frege’s distinction between objects and concepts, by contrast appears to be a distinction between two purely logical categories. To this extent I concur with Hylton. However, an important caveat must be added to 390 Ch. 4 Logic as the Universal Science I this statement. The Moorean, or metaphysical, conception of proposition has relatively little content in the absence of a further explanation of the constitution of propositions; and as we have also seen, the “vision” of logic that underlies Part I of the Principles is clouded, among other things, by Russell’s (intentional) indecision between Moorean and Peanist approaches to this problem. Applied to the present discussion, the point is that what the Russell of the Principles needs is a notion of proposition that can be put to use in the “logical analysis” of mathematical reasoning. This in turn means that for him the most important desideratum in the theory of propositions is to arrive at a conception of the composition of propositions that is sufficiently precise so as to make that conception genuinely useful in the analysis and reconstruction of mathematical reasoning. It is conceivable that a metaphysically driven conception of proposition that is not unlike the one that Russell got from Moore could be put to use in a definition of logic (as Hylton suggests in the above quotation). Whatever its content, however, such a definition would have been of little use to Russell, whose agenda in the Principles was primarily that of a working logician, rather than a philosopher of logic. In other words, Russell, like Frege before him, needed conceptual tools which would enable him to develop a notion of rigorous reasoning that could be applied in practice. In this respect the Moorean notion of proposition had little to offer: “proposition”, “constituent of a proposition” and “truth”, unless they receive further elucidation, are of little help in this respect. And recourse to “immediate acquaintance”, “non-sensuous perception” or “direct and immediate access” adds nothing, for these are really no more than words with little or no substance. Thus, it does not matter in the last analysis whether, given some sufficiently abstract perspective, one’s notion of proposition is seen as primarily metaphysical or logical; in order to do what he wanted to do, Russell needed a calculus (or, possibly, several calculi) in which rigorous mathematical reasonings could be reproduced.132

132 The fact remains that Russell’s logical practice was a good deal less Ch. 4 Logic as the Universal Science I 391

4.5.2.2.2 Further Remarks on Russell’s Conception of Calculus

There are good reasons, then, to think that the first rejoinder is inadequate as a response to the criticism advanced against (UA-1); the distinction between logic simpliciter and calculus for logic is relevant to Russell’s conception of logic for the reasons stated above, and observing this distinction enables us to maintain that the line of thought developed in (UA-1) does not exclude meta-perspective. The above criticism of (UA-1) can be supported by textual evidence. For it seems that Russell himself was sensitive to the distinction between logic simpliciter and calculus for logic. Consider the following quotation from his informal, or non-formal, presentation of “propositional calculus” in sections 14-19 of the Principles. What he has to say about his axioms in §17 strongly suggests that the term “propositional calculus” should be understood in the sense that we have just been delineating:

As regards our two indefinables [sc. material and formal implications], we require certain indemonstrable propositions, which hitherto I have rigorous than Frege’s (as Gödel (1944), among others, has pointed out). Unlike Hylton, I do not think that this difference is best explicated by referring to their putatively different conceptions of rigour: Frege’s syntactic and therefore essentially modern conception of rigour versus Russell’s metaphysical and therefore outdated and idiosyncratic conception. As the discussion of the concept of rigour in section 1.3.2 indicates, it is unlikely that there is any essential or doctrinal difference between the two in this respect. From this one need not conclude that Russell’s infamous sloppiness was simply a matter of carelessness (although it may have been that, too); another explanation – at least a partial one – is available here, to wit, that in practice Russell’s approach to rigour was closer to what a mathematician might find reasonable and appropriate, whereas the constraints that Frege imposed on genuine, rigorous proofs followed an altogether more stringent logical ideal; when thinking about this issue, it is instructive to reflect on what Frege had to say about Dedekind in the Preface to Grundlagen; according to Frege, the so-called proofs in Dedekind (1888) are not really proofs at all (Frege 1893, pp. 4-5). 392 Ch. 4 Logic as the Universal Science I

not succeeded in reducing to less than ten. Some indemonstrables there must be; and some propositions, such as the syllogism, must be of the number, since no demonstration is possible without them. But concerning others, it may be doubted whether they are indemonstrable or merely undemonstrated [...]. Thus the number of indemonstrable propositions may be capable of further reduction, and in regard to some of them I know of no grounds for regarding them as indemonstrable except that they have hitherto remained undemonstrated.

The point of view underlying this passage is not easily reconciled with Hylton’s interpretation of “external perspective”. Russell is clearly talking about the “principles of correct reasoning”, or principles which apply to any subject-matter and without which no demonstration is possible. A system of such principles, however, is nowhere given, but must be discovered and formulated, and to establish such a system (or “calculus”, as Russell calls it), one must engage in meta- theoretic reasoning. For example, given some suggested set of axioms for a logical calculus, there arises the question as to whether its axioms are independent or whether the chosen set of axioms allows further reduction. Although Russell’s attitude to independence proofs is certainly different from what is common today (see below, section 4.5.2.2.4), this does not vitiate the basic point: even on the universalist conception of logic, there is nothing illegitimate per se about “external perspective”and meta-theoretic reasoning; this presupposes, however, that the relevant terms are interpreted in the manner outlined above.

4.5.2.2.3 Reasoning about Reasoning

With the first rejoinder to (UA-1) dismissed, we are left with the second one. Here it is admitted that the distinction between logic simpliciter and calculus for logic does have an application to the universalist conception of logic. To this the further contention is added, however, that the universalist conception imposes non-trivial restrictions on the range of questions that a logician may legitimately ask about a Ch. 4 Logic as the Universal Science I 393 formal calculus for logic. According to this reply, the universalist logician reasons along the following lines. The principles of correct reasoning, i.e., the principles which constitute logic qua universal science, are presupposed – indeed, used – in reasoning about a calculus for logic. On the other hand, a calculus for logic is itself a codification of these very same principles of correct reasoning, or at least some of them; otherwise the calculus does not deserve to be called a calculus for logic. Hence, there can be no genuinely “external” perspective on logic, even when “logic” refers to a particular calculus in which deductive arguments can be reproduced, rather than logic qua universal science. This suggests the following modification of (UA-1), which we may call (UA-2):

(1) Logic comprises all principles of correct reasoning. (2) Hence, there can be no reasoning outside the one universal logic. (3) Reasoning about a calculus for logic presupposes/uses the principles of correct reasoning. (4) A calculus for logic is a codification of at least some principles of correct reasoning. (5) Hence, there can be no genuinely external perspective on a calculus for logic. (6) Semantic argumentation presupposes the possibility of adopting an external perspective on the system under consideration. (7) Hence, there can be no meta-theoretic semantic reasoning about a calculus for logic.

What specifically recommends (UA-2) as a reasonable explication of a part of the van Heijenoort interpretation is the fact that the argument does not presuppose, or at least can be read in a way that does not presuppose, the anachronistic sense of the term “external perspective” mentioned above – cf. section 4.3.3 – and discussed by Tappenden (1997). Recall that, according to Tappenden, there is a 394 Ch. 4 Logic as the Universal Science I considerable risk of anachronism in the attribution of an anti- semantical conclusion to a universalist logician (Tappenden discusses the case of Frege, but his point applies more generally); the inference from “no meta-perspective is available” to “semantics is impossible” is hardly one that can be taken for granted, as it is best associated with Tarski rather than the earliest contributors to mathematical logic. As a consequence, one should be cautious about attributing the view that something worthy of the name “semantics” requires an “external” perspective in this sense to someone like Russell, whose work on logic belongs firmly in the pre-Tarskian era. (UA-2) suggests, however, that the van Heijenoort interpretation need not attribute this understanding of “external perspective” to the universalist logician. Instead, the relevant sense of “externality” can be associated with the logocentric predicament that Harry Sheffer once identified as a major difficulty for any attempts to provide logic with a foundation:

[...] the attempt to formulate the foundations of logic is rendered ardu- ous [...] by a ‘logocentric’ predicament. In order to give an account of logic, we must presuppose and employ logic. (1926, p. 228; as quoted in Ricketts (1985, p. 3))

Logic has a legislative function wherever there is room for assertion, proof, reasoning, argumentation, etc.; this, we may agree, is both a traditional and very natural conception of logic. It is natural to say, furthermore, that logic in this sense – as the principles of correct reasoning – is presupposed in any attempt to provide a theoretical account of some subject-matter. When this is applied to logic itself, the following question arises: In what sense can logic itself be the subject of such an account? Logic delivers the principles which govern all valid reasoning; is there then available a standpoint from which these principles themselves can be explained? As Ricketts (1985, p. 3) points out, this question becomes all the more pressing, once foundational questions are clearly separated from questions concerning logical cognition: what needs elucidation, etc. is not our grasp of logical Ch. 4 Logic as the Universal Science I 395 principles, but the principles themselves – this, of course, is a characteristic anti-psychologistic move. The expressions “external perspective” and “external standpoint” are best understood in this way in the context of the van Heijenoort interpretation. Since any reasoning is subject to the laws of logic, these laws are fundamental to, and hence presupposed in, the provision of truth about any subject-matter; there can be no reasoning that is independent of these principles. The validity of the principles of correct reasoning are therefore also presupposed, when an attempt is made at establishing a calculus for logic, i.e., a codification – even a partial one – of these principles themselves, or when one wishes to prove that the calculus itself or some principle or rule featuring in it possesses this or that property. Since logic is in this way fundamental, it begins to look like there is no vantage point from which an account of logic itself could be given; that is, it begins to look like there can be no substantive metaperspective for logic.133

4.5.2.2.4 The Justification of Logic

It is clear that in the above formulation of (UA-2) the term “semantics” is being used in a rather special sense; to repeat, there is no reason, apart from the anachronistic one discussed by Tappenden, to think that an early universalist like Russell would have thought that semantic notions and semantic theorizing require an external perspective. (UA-2) is therefore not intended as a general argument to the effect that semantic notions and semantic theorizing are illegitimate; the argument is concerned with the more specific question whether semantic notions can be used for a certain purpose whose possibility presupposes something that deserves to be called an external perspective, to wit, the purpose of “giving an account of logic”. The question now arises: What does “giving an account of logic”

133 Cf. Weiner (1990, p. 227). 396 Ch. 4 Logic as the Universal Science I mean in this context? The suggestion that first comes to mind here is justification. To give an account of logic would be tantamount to giving a justification for the laws of logic. And it does seem that the advocates of the van Heijenoort interpretation do sometimes construe the issue in precisely these terms. Here are two examples, the first from Gary Kemp, the second one from Thomas Ricketts. Both passages are about Frege, but Kemp does make the parenthetical remark that the point applies to the early Russell, too:

[Frege] characterizes the theorems of logic, not as formulae or schemata true on all interpretations, but simply as the ‘most general laws’: whereas the special sciences represent the facts concerning some particular domain of reality – some particular range of objects and concepts – logic represents the most general and abstract facts, ones which hold for all objects and concepts whatsoever (Russell advances a similar view in The Principles of Mathematics). Logic thus serves as the prescriptive ‘laws of thought’, not as a universally applicable method of ensuring validity, but as a set of universally constraining facts. It follows that there is no external perspective from which the deduction of logical laws form others can be thought of as being justified. Every judgement needed in the establishment of a theorem of logic must itself be expressed as a line in the proof, not as part of a separate argument to the effect that some particular mode of inference is correct. (Kemp 1998, p. 222; italics added)

Even apart from its use of a truth-predicate, Frege would find the attempt to prove his formalism sound to be pointless. Such a proof could achieve scientific status only via formalization inside the framework provided by the formulation of logic it proves sound. The resulting circularity would, in Frege’s eyes, vitiate the proof as any sort of justification for the formalism.” (Ricketts 1995, p. 136; italics added)

In both of these passages the point is made that a universalist must reject the idea of justifying logic. Although semantics is not mentioned by name in the first quotation, it is a natural assumption that what is at stake is the possibility of a semantic justification of logic; this is made explicit in the second quotation. The question, then, concerns the prospects of an inferential justification of logic which Ch. 4 Logic as the Universal Science I 397 takes the form of an argument to the effect that a given logical rule is necessarily truth-preserving, or an axiom true, or a proof that a given formalism is sound (that every deductively valid argument is semantically valid). Both Kemp and Ricketts point to a well-known difficulty with such arguments and proofs. The justification of logic is bound to involve some reasoning, and the correctness of a reasoning is dependent upon the correctness of the laws and rules in which it is grounded. What is at stake in the justification of logic, however, is precisely the correctness of the principles underlying correct deductive reasoning. Hence, it seems that a putative inferential justification of logic inevitably ends up being circular. It is a familiar point about such arguments that their circularity is not of the immediately damaging type, where the conclusion to be established is found among the premises for that conclusion. In other words, when it comes to the justification of logic – to keep things simple, we may concentrate on the justification of a logical rule – the charge is not that a justificatory argument has among its premises the claim that the rule in question is valid. Rather, the circularity in question is of the kind that is variously known as “pragmatic circularity” (Dummett 1991, p 202), “rule-circularity” (Boghossian 2000, p.232), or “Cartesian Circle” (Heck 2007, p. 29). Dummett explains the difference between the two types of circularity as follows:

We have some argument that purports to arrive at the conclusion that such-and-such a logical law is valid; and the charge is not that this argument must include among its premisses the statement that that logical law is valid, but only that at least one of the inferential steps in the argument must be taken in accordance with that law. (1991, p. 202)

As Dummett (id., p. 201) points out, insofar as we are concerned with the justification of a single rule, the general claim that every semantic justification of a logical law must appeal to that very same law in the course of the justificatory argument is too strong. It is not the case, that is, that an argument for any logical rule will inevitably refer to the 398 Ch. 4 Logic as the Universal Science I rule itself; there could be justificatory arguments for some logical rules which only makes use of elementary logical laws. Taking this into account, the claim would be that the fundamental or basic laws of logic cannot be justified via deductive arguments (and hence cannot be justified at all, if we assume that there are no serious alternatives to inferential justification). The reason for this would be, precisely, that such arguments are necessarily ‘rule-circular’; being an argument, the justification involves inferential steps whose legitimacy presupposes the validity of the law for which the argument is supposed to act as a warrant.134 The distinction between pragmatic circularity and gross circularity promises some leeway for a philosopher who wishes to defend the idea of justificatory argument. It will be admitted that no such argument can have persuasive force; a logical sceptic, or a person who genuinely and sincerely entertains doubts about the validity of a fundamental law of logic, will not be convinced of the validity of the law by a deductive argument. But the justification of a logical law need not, and typically is not, presented with an intention of convincing a logical sceptic. More likely, the argument is meant to constitute an explanation of why its conclusion is true. This type of argument may be dubbed explanatory argument. Since justification, so understood, is not intended to persuade but to help someone who is seeking an understanding of its source or ground, its circularity is not harmful. These considerations indicate that (UA-2) in fact only succeeds in opening up a rather narrow perspective on the universalist conception of logic. The point underlying (UA-2) is that a reasoning can

134 This is easiest to illustrate with modus ponens. A justification of modus ponens would presumably start with the identification of a condition which this rule has to fulfil in order to be valid; since we are dealing with semantic justification, the condition would naturally refer to the truth-table definition of the material conditional. This yields the conditional, ‘if such- and-such a condition is fulfilled, then modus ponens is a valid form of inference’. Next, it is established that modus ponens does, indeed, fulfil the condition mentioned in the antecedent of the conditional. Finally, it is concluded, by modus ponens, that modus ponens is a valid form of inference. Ch. 4 Logic as the Universal Science I 399 achieve a status as objective and scientific only by being reproduced in a calculus emulating universal logic (this is how Ricketts puts in the above quotation). In other words, the universalist is construed as holding that an inference expressed in an ordinary language or its extension can be converted into a genuine proof only by reproducing it in the universalist calculus. In judging the correctness of a reasoning, the ultimate reference point is thus the correctness of a law or rule featuring in that calculus. There is nothing more fundamental – like facts about models or truth conditions – to which the correctness of a logical law could be reduced. Semantic facts cannot therefore be the ultimate source of logical correctness or validity, according to the universalist. It appears, however, that the model-theoretic logician is not in a significantly different position, in this respect, from the universalist logician. He does propose model-theoretic explications for a number of crucial logical concepts; in particular, he suggests that the concept of logical consequence should be understood semantically, with the help of models. This approach, however, cannot be thought of as achieving a reduction of the target-concept to something more basic. One way to argue for this conclusion would be from the recognition, as in Prawitz (1974, pp. 67-68), that the model-theoretician’s procedure in fact amounts to, first, giving a definition of logical consequence in semantic terms, and then declaring that a particular sentence follows from another sentence or sentences, only if this follows logically from his definition. Again, this conclusion is to be modified by pointing out that this does not render the model-theoretician’s procedure methodologically defective, since the point behind the procedure is not to give a justification via reduction. But the conclusion does show that, when it comes to the justification of a logical law in some fundamental sense, there is no difference between the mode-theoretic and universal logicians. Studying their respective attitudes towards the possibility of justificatory arguments reveals no significant difference between the universalist and model-theoretic logician. The universalist’s claim is that 400 Ch. 4 Logic as the Universal Science I a calculus ‘modelling’ the universal logic, or some particular feature of that calculus, cannot be given a justification, because any attempted justification – a deductive argument – should ideally be formulated in the calculus itself. The model-theoretic logician has available to himself the distinction between object- and meta-levels and is therefore in a position to argue that reasoning about a logical calculus is itself an external matter. Nevertheless, when it comes to justification, the model-theoretic logician’s position is not essentially different from that of the universalist; the question of justification now reappears as one concerning the reasonings belonging to the meta-level. We are thus left with explanatory, as opposed to justificatory, arguments and the provisional conclusion that (UA-2) leaves a number of meta-theoretical projects unscathed, namely those that are best characterized in terms of explanation, rather than justification. Even after recognizing the validity of this distinction and its relevance for an assessment of the merits of (UA-2), an advocate of the van Hei- jenoort interpretation may still press the point that a genuine and profound disagreement remains between the universalist and model- theoretic logician, when it comes to the possibility and extent of such explanatory arguments. For an explication of the nature of this potential disagreement, however, we should turn to the second argument for the universalist conclusion, namely (FCA), the argument from fixed content.

4.5.2.3 What is Wrong with the Fixed Content Argument

4.5.2.3.1 Justification and Semantic Explanation

I begin this section by repeating the fixed-content argument (FCA). The following formulation, however, is somewhat different from the one given in the previous section: Ch. 4 Logic as the Universal Science I 401

(1) It follows from the thesis of fixed content that the propositions of logic have content and, therefore, truth-value, independently of interpretation. (2) Semantic reasoning presupposes the possibility of alternative interpretations. (3) Therefore, the thesis of fixed content excludes semantics, or semantic explanation.

I present (FCA) in this simplified and modified form in order to emphasize the following two points. First, the conclusion is formulated in terms of explanation. As we have seen, this provides us with a reasonable way of understanding what “semantics” means in the context of the universalist conception of logic and the differences between that conception and the model-theoretic one; to be sure, ‘explanation’ requires further clarification, but we will turn to issue in a moment. Secondly, the question concerning the possibility and extent of semantic explanation is closely related to the notion of interpretation. Hence, the issue whether semantics in the sense of semantic explanation is possible is very close to, if not identical with, the question whether interpretation (dis-interpretation and reinterpretation) is possible. Looking at (FCA) promises us a reasonably clear-cut set of issues with respect to which the two conceptions of logic might turn out to differ from each other. From the model-theoretic standpoint, these two points are very closely related. In the previous section it was argued that there are good reasons to accept the common view according to which any attempted justification of a law of logic is necessarily circular. How- ever, as Heck (2007, p. 41) points out, this claim depends upon the assumption that the laws whose justification is in question are the thoughts or propositions expressed by certain sentences. But there is no circularity if the justification concerns, not thoughts or propositions, but the sentences that we use to express these thoughts or propositions. For instance, there is no circularity in arguing that, on a certain interpretation for the expressions occurring in it, the sentence ‘A ¬A’ that 402 Ch. 4 Logic as the Universal Science I in fact expresses a law of logic, is true. Heck points out further that semantic theories frequently have this kind of purpose. Given a syntactic specification of a language of logic, including a specification of a number of axioms and a proof- procedure, a semantic interpretation assigns references to primitive expressions and provides rules which determine the references of compound expressions on the basis of their constituent expressions, leading to a definition of truth relative to an interpretation. With the help of this semantics, arguments can then be given to show that the axioms of the system in question are true and its rules of inference truth-preserving. We may even revert to our earlier terminology and say that this procedure constitutes a justification based on semantic explanation. Clearly, justification in this sense – it could be called semantic justification – is non-circular. The point of semantic justification is not to demonstrate that certain propositions or thoughts are true or that certain transitions from propositions or thoughts to other propositions or thoughts are truth-preserving, but merely that certain sentences, i.e., strings of symbols interpreted in a particular way, that are treated as axioms are true and that certain transformation rules, or rules that license transitions from sentences to sentences, are truth- preserving. We accept such demonstrations as correct (if we do), because we accept as correct certain claims that belong to the metalanguage, or the language in which the interpretation is given and the justificatory argument carried out. This presents no problem to the project at hand, since the point is merely to demonstrate that certain object-language sentences and inferences have certain desirable properties. But if this is the sense of “semantic explanation” that is relevant to (FCA), it is not immediately clear that there is much difference between the model-theoretic and universalist logician in this respect. For it appears that on this reading of “semantic explanation” the inference from the fixed-content thesis to the anti-semantical conclusion (“semantic explanation is impossible”) remains somewhat ill- motivated. Reading the secondary literature, one often encounters the Ch. 4 Logic as the Universal Science I 403 idea that the universalist’s anti-semantical stance is grounded in the conception of logic as language. On this view, the universalist regards logic as a meaningful formalism;135 that is, logic is something like a language in the ordinary sense, albeit a language from which the ambiguities and other blemishes of natural languages have been eliminated and which is formulated with sufficient precision to allow the application of a mechanical proof-procedure. Since the language of logic is already fully interpreted, it is argued, no role is left for a semantic investigation of the formalism. This account, however, fails to explain why a language that is fully interpreted could not also be treated as a calculus.136 Furthermore, that a universalist can treat logic in this way appears to be more than just an abstract possibility. For instance, there are good reasons to think that this is precisely how we should think of Frege’s logic.137

4.5.2.3.2 Frege on the Semantic Justification of Logic

Frege thought that the ultimate subject-matter of logic consists in something non-linguistic, namely, the entities that he referred to as ‘thoughts’ in his mature semantics: logical relations, according to Frege, are in the most fundamental sense relations between thoughts. Nevertheless, it is also fairly clear that he did concern himself with semantic theory in Part I of his Grundgesetze: the early parts of that work are concerned, among other things, with the semantic justification of his concept-script. That is, he explains the primitive signs of his system and proceeds thence to explain, as Burge (1998b, p. 320) puts it, “the intended sense of his formulae by giving their truth-

135 I borrow this expression from Heck (2007, p. 43). 136 As is pointed out by Tappenden (1997, pp. 241-2). 137 See Stanley (1995), Tappenden (1997) and Heck (2007) for this line of thought; see also Burge (1998b), where the perspective is somewhat different, but the point is developed in considerable detail. 404 Ch. 4 Logic as the Universal Science I conditions”. The point of this exercise is precisely to justify the formulas of the logical language under scrutiny – i.e., Frege’s concept-script – by “showing that they express logical truths and valid inferences, which are antecedently understood to be self-evident” (ibid.) For instance, in §14 of Grundgesetze, entitled “First Method of In- ference”, Frege writes: “From the propositions ‘»î (ƅȺƄ)’ and ‘»î ƅ’ we may infer ‘»îƄ’; for if Ƅ were not the True, then since ƅ is the true (ƅȺƄ) would be the False.’ As this quotation indicates – and as Burge (1998b, p. 320) points out – Frege’s “method of inference” is a rule for moving from sentences to sentences. It is true that Frege’s explanation is not entirely free of use-mention confusion: he begins by speaking about linguistic entities (“propositions”; Sätze), but switches immediately to the material mode. The explanation is thus not explicitly semantical: Frege does not speak of what the premises and conclusion, i.e., certain sentences, denote but of certain objects’ being the True and the False (cf. Heck 2007, p. 49). This switching between the formal and material modes, however, appears to be of little significance. As the above quotation from Burge indicates, there is a more fundamental level at which logical theory operates, according to Frege. This is the level of thoughts. And there is certainly no justifying the axioms and rules of inference in this fundamental sense; no amount of semantic reasoning, or reasoning that is essentially about sentences, expressions, etc., can show anything worthwhile about axioms qua thoughts or inferences qua transitions from, as Frege would put it, asserted thoughts to asserted thoughts; Frege simply assumes a family of true laws and correct rules that pertain to thoughts (hence the reference in the quotation from Burge to self-evidence), and these are then available as the ultimate reference point when one proceeds to justify this or that feature of the logical language. But it is quite clear that these explanations are not intended to apply to the fundamental level of thoughts, but concern the formulae of Begriffsschrift; thus, what Frege is doing in Part I of Grundgesetze can be described as a case of semantic justification. Ch. 4 Logic as the Universal Science I 405

4.5.2.3.3 Calculus for Logic and the Science of Logic

It helps to sort out the present interpretative issues, if we relate them to the distinction between a calculus for logic and logic as science that we draw in section 4.5.2.2. The basic picture that emerges from our discussion is simply as follows:

Calculus for logic is a Logic as universal model science Sentences of p is Thoughts (or other Semantic justification pre- non-linguistic items) p sup- p Semantic explanation posed by Basic norms for correct reasoning

Fig. 4.3 The universalist conception of logic

Frege’s case provides a good illustration of how important it is to keep this distinction in mind when one considers the universalist conception of logic. Observing the distinction enables us to see that the universalist conception is perfectly compatible with semantic reasoning, and more generally with the so-called “meta-theoretic” perspective. One can even argue that meta-theoretic reasoning has a very clear function in that conception; for reasoning about a calculus can be used to argue that the calculus does in fact fulfil a set of conditions that it should fulfil, if it is to count as a calculus for logic. We need to proceed with caution here, however. Even if our discussion of what is and what is not involved in or implied by the universalist conception of logic is backed up by a reference to an existing logical practice, like that of Frege’s, the fact remains that the conception thus outlined – the conception summarised in figure 4.3 – is an abstract ideal type. Once we turn to consider Russell and his concep- 406 Ch. 4 Logic as the Universal Science I tion of logic in more detail, we cannot assume without further ado that the conceptual possibilities inherent in the universalist conception were recognized as such by him. Whether and to what extent Russell’s views comply with what is proposed by the above figure, is thus a question that requires separate investigation. I have already argued that the distinction between logic as a calculus and logic as science was relevant to Russell’s thinking about logic.138 Like Frege, he needed a calculus for logic in which mathematical reasonings could be represented or reconstructed in a rigorous manner. It is only in this way, by actually deriving mathematics, or some portion thereof, from its logical foundation that a thesis of logicism could be proved correct. However, a calculus for logic, or logical framework in which the derivation can be carried out, is nowhere given, but must be discovered and formulated. To establish such a calculus, a certain amount of meta-theoretical reasoning is necessary. Russell’s discussion of logic in the Principles, Chapter II, is a discussion of logical calculi, rather than logic as a science. After a number of preliminary remarks on the nature of logic, he proceeds to give, in §§14-19, an informal, or non-formal, presentation of what he calls “propositional calculus”. This is followed by similar expositions of the calculus of classes (§§20-26) and the calculus of relations (§§27-30). Of these three calculi, the first, which studies the relation of material implication, is the most fundamental or elementary part of mathematical logic: it constitutes, as he would later put it, “the first chapter of the deduction of pure mathematics from its logical foundations” (1906d, p. 159). That is, propositional calculus is concerned with deduction itself, i.e., with “how one proposition can be inferred from another” (ibid.). In one sense, then, Russell’s calculus of propositions is concerned with the principles of correct reasoning; that they are fundamental in nature is shown by the fact that, without them, no “demonstration is possible” (§17). When Russell discusses symbolic or formal logic in these sections he is nevertheless concerned with a

138 Cf. section 4.5.2.2.1. Ch. 4 Logic as the Universal Science I 407

“technical study” or “formal development” of the subject (§12), and here convenience is always a relevant consideration; for instance, the propositional calculus is conveniently developed as a theory about material implication, given its close association – at least when it is coupled with the notion of formal implication – with inference. Russell’s calculi thus deal with the principles of correct reasoning. They deal with principles that help us to adjudicate upon whether a particular piece of deductive reasoning is correct – and, indeed, constitute the ultimate reference point, whenever the issue of deductive correctness arises. In order to set up a satisfactory system of such principles, one must address a number of meta-theoretical issues. Russell is not unaware of this requirement. And this point is not undermined by the recognition that, although he is concerned with a technical development of the subject, the presentation of symbolic logic in chapter II of the Principles leaves much to be desired from a technical point of view.139

4.5.2.3.4 Russellian Metatheory?

A meta-theoretical perspective on a logical calculus is nevertheless not wholly absent from the Principles. That such a perspective is relevant to Russell’s logicist enterprise is shown very clearly by the comments inserted at the beginning of a somewhat later work, namely the long 1906-paper, The Theory of Implication, which is his first truly systematic presentation of a significant part of deductive logic:

139 It should be borne in mind, though, that the Principles is intended to serve as no more than a prolegomenon to logicism; to compare the work with Frege’s Grundgesetze, for instance, would be quite unfair to Russell. The level at which the Principles operates is not the level of logical detail – which, at any rate, would have been far beyond the horizon of an average English- speaking philosopher at the beginning of the 20th century – but the level at which a general argument is conducted. 408 Ch. 4 Logic as the Universal Science I

When a proposition q is a consequence of a proposition p, we say that p implies q. Thus deduction depends upon the relation of implication, and every deductive system must contain among its premisses as many of the properties of implication as are necessary to legitimate the ordinary procedure of deduction. In the present article, certain propositions concerning implication will be stated as premisses, and it will be shown that they are sufficient for all common forms of inference. It will not be shown that they are all necessary, and it is probable that the number of them might be diminished. All that is affirmed concerning the premisses is (1) that they are true, (2) that they are sufficient for the theory of deduction, (3) that I do not know how to diminish their number. But with regard to (2), there must always be some element of doubt, since it is hard to be sure that one never uses some principle unconsciously. The habit of being rigidly guided by formal symbolic rules is a safeguard against uncon- scious assumptions; but even this safeguard is not always adequate. (1906d, pp. 159-160)

The three features that Russell mentions in this passage may be labelled, respectively, soundness, completeness and independence. Of these, the third is clearly the least important; a failure in independence is, so to speak, a mere aesthetic blemish.140 By contrast, soundness and completeness are crucial, if the calculus is to serve its purpose. Since the calculus is concerned with deduction, it should track consequence, which means that its axioms should be true and its rules of inference truth- preserving; otherwise it cannot be said to consist in “principles of correct reasoning”. Also, the calculus should be capable of doing everything that we, intuitively, want it to do; as Russell puts it in the above quotation, the calculus that is concerned with deduction itself should suffice “for all common forms of inference”. There are thus good reasons to maintain, as against Hylton’s explicit statement to the contrary, that meta-theoretical ideas were not “foreign to Russell’s conception of logic” (Hylton 1990a, p. 202); at any rate, they were not foreign if by this one means that he could not have as much as raised meta-theoretical questions concerning a logical calculus.

140 “Redundancy is not a logical error, but merely a defect of what may be called style” (Russell 1903a, §122). Ch. 4 Logic as the Universal Science I 409

An advocate of the van Heijenoort interpretation has the following response available. It may be admitted that meta-theoretical issues as such are not, strictly speaking, impossible for a universalist like Rus- sell. Nevertheless, the way the universalist addresses them is different from that of the model-theoretic logician’s. This can be illustrated by considering the notion of completeness. There are passages from universalists that are undeniably concerned with completeness; we have already seen Russell alluding to it (a theory of deduction, he explains, should contain as many primitive propositions or axioms about the relation of implication “as are necessary to legitimate the ordinary procedure of deduction”).141 However, it may be argued that such passages confirm rather than refute the van Heijenoort interpretation.142 The contention is that recognition of meta-theoretical questions does not lead the universalist to take “any jump to a metalevel” (Floyd 1997, p. 150); rather, the universalist assumes an experimental attitude towards meta-theoretical issues.143 Applied to completeness, this means that the universalist does not seek a global meta-theoretical proof that a calculus has this property; he operates, instead, within the calculus, trying to produce proofs for everything that should be provable, and modifying, if necessary, the system in order to accommodate the desired results (such modifications would, of course, take place within a number of constraints). Russell would later dub this sort of reasoning “inductive”, arguing that it plays a major role in the epistemology of mathematics and logic.144

141 Similar passages are also found in Frege: see Frege (1879), §13 and (1897), p. 235. 142 According to Floyd (1997, p. 150), Burton Dreben argued for this, in Frege’s case, in unpublished lectures. Ricketts (1997) and Allnes (1998, §11) defend a similar interpretation of Frege. 143 The term “experimental” was used in this sense by van Heijenoort (1967, p. 327), who says he took it over from Herbrand. 144 Russell (1907b) is an extended discussion of this kind of inductive reasoning. The “regressive method”, as he calls it, is not only the method of 410 Ch. 4 Logic as the Universal Science I

There is thus no denying the difference between the universalist’s inductive approach to completeness and the model-theoretic logician’s meta-theoretical perspective. Indeed, in light of this difference, it might be preferable to call the universalist version of completeness “comprehensiveness”, and reserve the term “completeness” for its customary usage.145 But we should exercise some care in identifying the reasons for this evident difference between comprehensiveness that is to be established inductively (derive as many theorems as you can from these axioms and rules of inference and see if they can give you everything you would like to have!) and completeness (are the axioms and rules of inference that comprise this system capable of delivering every formula that is intuitively valid?) We have already seen that universality in itself is not enough to exclude meta-theoretic reasoning, and this point applies to completeness as well. Hylton explains Russell’s attitude towards completeness as follows:

Consider [...] the question of the completeness of a system of logic, which is so natural for us. This question relies upon the idea that we have, independently of the logical system, a criterion of what the system ought to be able to do, so it relies upon the essentially meta-theoretic notion of an interpretation, and of truth in all interpretations. These meta- theoretic ideas, however, are foreign to Russell’s conception of logic; the question of the completeness of a system simply could not arise for him. Logic for him was not a system, or a formalism, which might or might not capture what we take to be the logically valid body of schemata [...] (1990a, p. 202).

Hylton’s explanation can be seen to be less than compelling, once we discovery in mathematical logic, but offers as well a partial justification for one’s belief in the truth of the axioms from which results can be deduced that are, comparatively speaking, more familiar, than their ultimate “logical premises”. See Irvine (1989) and Coffa (1991, pp. 120-2)) for further discussion of Russell’s inductive logicism. 145 The term “comprehensiveness” is adopted from Ricketts (1997, p. 148). Ch. 4 Logic as the Universal Science I 411 remind ourselves of the fact that for a universalist, the term “logic” refers not only to the principles of correct reasoning but also to a calculus which purports to codify at least some of these principles. Assuming with Frege and Russell and other logical realists that there are such system-transcendent principles, it evidently makes little sense to say, let alone prescribe, that they should do this or that; as Frege and Russell both emphasized, our cognitive relation to them is one of recognition rather than stipulation.146 When it comes to a system of logic or logical calculus, however, the situation is different. In this case there is, even for a logical realist, an element of stipulation involved. Evidently, our stipulations do not concern the truth of axioms or the validity of rules of inferences; they concern the framework in which these axioms and rules are presented. But the principles of correct reasoning (or principles that one takes to contribute to the delineation of correct reasoning) are operative here, too, for reflection on these principles provides one “criterion of what the system ought to be able to do”. In other words, meta-theoretical questions are questions about a proposed logical system or framework. Hence, the universalist conception is not in itself sufficient to render illegitimate such questions as completeness, as Hylton suggests in the above quotation. On the other hand, if Hylton’s point is merely that Russell lacked a notion of completeness as distinct from comprehensiveness, then he is probably correct. In that case, however, Hylton still owes us an

146 According to Frege, the laws of logic are the most general laws of truth (1918-19a, p. 351), i.e. “the guiding principles for thought in the at- tainment of truth” (1893, p. xv); since being true is independent of being taken to be true, it follows that the laws of logic must be independent of what we take them to be. Russell evidently shared this sentiment with Frege; it is a simple corollary of their ubiquitous anti-psychologism. For instance, in the Principles, §427, Russell argues that numbers are objective entities, and that “Arithmetic must be discovered in just the same sense in which Colum- bus discovered the West Indies”; evidently, if arithmetical truth ultimately reduces to logical truth, the latter must enjoy the same status as the former. Hence the laws of logic must be independent of us. 412 Ch. 4 Logic as the Universal Science I explanation as to why this should have been so: it is not enough to cite the universalist conception of logic, maintaining that it somehow prevented Russell from having a criterion of what a system of logic should be capable of doing. If we want to identify the reason or reasons for Russell’s neglect of completeness (as distinct from comprehensiveness), we should focus on the notion of interpretation that underlies our notion of semantic completeness. We are thus back to (UA-2) and the thesis of fixed content; to the extent that Russell accepted this thesis, he could not have had the modern concept of interpretation, either, or some notion sufficiently similar to it. And without some notion of interpretation, he could not have formulated our question of completeness. Let us consider soundness next. As the quotation from The Theory of Implication shows, Russell is well aware that this requirement must be met. This much can also be gleaned from the Principles, even if he is less explicit about it. The first thing to be noticed about the relevant passages in the Principles is this. Though Russell was perfectly capable of drawing the distinction between logical rules and logical principles, or axioms, he tends to ignore it in his logical practice. Thus the question of soundness appears to him as the question whether the “principles of deduction” are true, as in the quotation from The Theory of Implication and the Principles, §17. In the latter passage – to which we shall return below when we discuss the issue of independence – Russell maintains that all the axioms of the propositional calculus are principles of deduction and, as such, must be true, for otherwise the consequences that appear to follow from their use would not really follow. Among the ten axioms that constitute Russell’s calculus there is one whose status is exceptional. This principle is axiom (4), which says that “[a] true hypothesis in an implication may be dropped and the consequent asserted”.147 This principle is the Russellian equivalent of modus ponens or the rule of detachment. As Russell explains in the Principles, §38, this axiom-cum-rule is what is used whenever some-

147 Cf. *2.1 of Russell (1906d). Ch. 4 Logic as the Universal Science I 413 thing is proved: unlike material implication, it enables us actually to assert a proposition that is implied by another proposition or propositions. Russell makes it clear that this principle is quite different from the other axioms of his calculus; according to him, the difference manifests itself in the fact that the principle cannot be formalized or stated in symbols. If we did not recognize such extra-systematic principle, we could not deduce or infer anything (this is the principal lesson that Russell derives from Lewis Carroll’s “What the Tortoise said to Achilles”; for further discussion, see section 5.9.3). This shows, as Landini (1998, p. 34) puts it, that “Russell did have a sense of what we call the difference between meta-language and object-language”. For our present purposes, the point is this; Russell’s undifferentiated requirement that the principles of a logical calculus must be true does not show that he neglected to state this condition for the rules of inference; his use of “logical principle” is more liberal than ours, since it covers not only axioms (primitive propositions) but rules of inference as well. The second, and in this context more substantial, point is that Russell’s awareness of the soundness requirement does not really lead him to put forward a semantic argument for the conclusion that the principles of his calculus have this property; unlike the Frege of Grundgesetze, Russell does not really offer a semantic justification based on semantic explanation. To the extent that he addresses the question of soundness in the Principles, he is content with pointing out that his axioms are all self-evident. And even this is mentioned only as an aside, with no elaboration on the relevant notion of self- evidence.148

148 In §18, where Russell enumerates the ten axioms constituting the calculus of propositions, he gives, as axiom (10), the principle of reduction: if p and q are propositions, then “‘p implies q’ implies p” implies p. He admits that this principle has “less self-evidence” than the other axioms, but he argues that we can easily convince ourselves of its truth, once we remember the definition of material implication: ‘If we remember that “p implies q” is equivalent to “q or not-p”, we can easily convince ourselves that the above 414 Ch. 4 Logic as the Universal Science I

It is a plausible contention, furthermore, that Russell would not have really wanted to insist that self-evidence is a necessary feature of axioms; an axiom that has little self-evidence, or one that has less self- evidence than some other potential axiom, may nevertheless be included among the primitive propositions of a calculus for reasons of simplicity or because it complies better with the general character of the calculus. For instance, Russell argues that the principle of reduction (see the previous footnote) is better suited as an axiom than excluded middle or double negation, because, although the latter two appear to have more self-evidence, the principle of reduction is explicitly concerned with implication, which is the proper subject- matter of the propositional calculus. It seems, then, that at least in the Principles Russell’s perspective on the logical calculus is not really semantic at all; unlike Frege, Russell does not provide a semantic explanation-cum-justification for his axioms. That is, he does not present arguments to the effect that, under the intended interpretation, or the interpretation that makes the calculus a calculus for logic, this or that sentence that in fact expresses an axiom is true.

4.5.3 The True Source of Russell’s Anti-Semanticism

Our discussion of completeness and soundness points in the following direction. What distinguishes Russell’s conception of logic from some more modern and essentially semantic one is not that the former excludes meta-theoretical questions while the latter leads at once to a number of such questions; the real difference is that these questions take different forms on each conception. Semantic justification and semantic explanation in the sense discussed above arise naturally in the context of the modern concept, whereas the semantic perspective appears to be something that the early Russell did not consider. Fur- principle is true; for “‘p implies q’ implies p” is equivalent to “p or the denial of ‘q or not-p,’” i.e. to “p or ‘p and not-q,’” i.e. to p.’ (§17) Ch. 4 Logic as the Universal Science I 415 thermore, as we have seen, there is a difference in this respect between Russell and Frege, too; hence semantic justification and explanation do not presuppose a model-theoretic conception of logic. This anti-semantical stance (or, to use a less loaded term, relative neglect of semantic considerations) does not arise from the universal conception of logic itself. Rather, its source is to be found in the early Rus- sell’s notion of proposition. Russell agreed with Frege that logic in the most fundamental sense is not a matter of language, but applies first and foremost to non-linguistic entities; in Russell’s case these are propositions construed as worldly entities. If, now, we think of semantics as something like a systematic theory of the relationship that holds between this or that language and the world, it follows at once from an assumption like the one that is involved in Russell’s notion of proposition that a semantic perspective cannot be germane to logic. To put the point another way, if a semantic study of logic has something to do with linguistic meaning, then, since the latter is superfluous, so is the former. As we have already seen, the thesis hat linguistic meaning is irrelevant to logic is a direct corollary to how he conceived of propositions (cf. section 4.4.1): since propositions are as a rule non-linguistic entities, they do not contain words but the entities indicate by words; therefore, meaning in the sense that words have meaning is irrelevant in logic.149 The so-called semantic conception of truth, which in its explicit form originates with Tarski, is an important ingredient in our thinking about logic. On this conception, truth and falsehood are applied primarily to linguistic entities. The truth-value of a sentence is the outcome of fixing two parameters: the semantic parameter (what does this sentence mean?) and the worldly parameter (are things as the sentence says they are?). Thus, Tarski – like any sensible theorist of truth who construes truth as a property of sentences – thinks that truth attaches to sentences plus their meanings. Russell, by contrast, seeks to bypass the first parameter, assigning truth and falsity directly

149 This is stated in §51 of the Principles; the direct quotation was given in section 4.4.1. 416 Ch. 4 Logic as the Universal Science I to propositions.150 Hence, for Russell, the entities that are true and false are so on their own right, independently of any fixing of a semantic parameter or semantic interpretation. What makes Russell’s logical framework anti-semantic is not the fact that he has no concept of truth; he has it and it has an important, though largely implicit, role to play.151 This shows that the possibility of semantics does not depend upon the presence of the concept of truth; we should say, rather, that it depends upon the availability of the concept of truth-condition.152 It is this concept of truth-condition that is essential to the semantic perspective, and since the early Russell has no use for the concept, he lacks,

150 The argument underlying this move from sentences to propositions is often formulated as follows. Truth and falsity attach to what we say, but not in virtue of the audible shape of our utterance but in virtue of its meaning; this is how Dummett (1981, pp. 37-8) puts it. On the other had, it is natural to construe propositions as meanings of sentences of a particular kind; this move, we have seen, is also present in the Principles. Therefore, truth in the fundamental sense is a property of propositions, rather than the physical tokens that are the carriers of information. According to Dummett (ibid.), the weak point in this argument is its neglect of meaning. Assigning truth and falsity directly to propositions, the classical theorists of truth failed to forge a connection between truth and meaning; that is, they failed to explain how truth attaches to what we say. As for the underlying argument itself, I am inclined to think that in the case of such philosophers as the early Russell, their actual reason for working with some notion of proposition is to be found in a number of assumptions about the independence of truth, rather than assumptions about (linguistic) meaning. This line of thought is developed explicitly in Bolzano (1837, §25), where the author explains that truth is independent in a number of ways: it is independent of the knowing subject, of location and time of utterance as well as of changes in the circumstances of the world (I am here following the exposition in Rojszczak (2005, sec. 7.1)). Since truth is in this way independent, a theory relating truth to meaning cannot be germane to theory of truth, although it could well turn out to be relevant to a theory of meaning. That is to say, truth in the fundamental sense is not a semantic notion at all. 151 See above, section 4.4.6.4, and below, section 4.5.7. 152 On this point, see Stenius (1984, pp. 228-9). Ch. 4 Logic as the Universal Science I 417 to that extent, a semantic perspective on logic. One could also argue in the following way. The constituents of Russellian propositions are worldly entities; Socrates, the person, rather than the proper name “Socrates” or the mental item that a speaker is said to associate with that name, occurs as a constituent in the proposition that Socrates is mortal. Hence, the procedure of interpretation – dis-interpretation, reinterpretation – makes little, if any, sense, when applied to these entities. To the extent that this view about truth-bearers underlies Russell’s conception of logic, it provides a simple and straightforward explanation of his anti-semanticism. To put the point in terms of our previous discussion, it appears that (FCA), the argument from fixed content, and with it, Russell’s neglect of semantics, follows as a simple corollary from his construal of propositions as non-linguistic entities: propositions have content independently of interpretation; hence a semantic perspective cannot be germane to logic. This line of thought is certainly attractive, as it provides what appears to be a rather simple and convincing argument for attributing something like (FCA) to Russell. We must nevertheless ascertain whether, and if so, to what extent, this conclusion is justified.

4.5.4 Russell on Alternative Interpretations

Given the above argument, one would naturally think that, insofar as Russell considers semantic interpretation, or something sufficiently similar to it, he would be critical of such ideas. The closest he comes to a criticism of a notion of alternative interpretations is an argument given in §17 of the Principles. It purports to explain why the independence of logical axioms cannot be established by what he calls “recognized methods”:

We require, then, in the propositional calculus, no indefinables except the two kinds of implication [...]. As regards our two indefinables, we require certain indemonstrable propositions, which hitherto I have not 418 Ch. 4 Logic as the Universal Science I

succeeded in reducing to less than ten. Some indemonstrables there must be; and some propositions, such as the syllogism must be of the number, since no demonstration is possible without them. But concerning others, it may be doubted whether they are indemonstrable or merely undemonstrated; and it should be observed that the method of supposing an axiom false, and deducing the consequences of this assumption, which has been found admirable in such cases as the axiom of parallels, is here not universally available. For all our axioms are principles of deduction; and if they are true, the consequences which appear to follow from the employment of an opposite principle will not really follow, so that arguments from the supposition of the falsity of an axiom are here subject to special fallacies.

At the beginning of this quotation Russell purports to establish a conclusion that is familiar from our earlier discussion of justification of logic. Since the validity of demonstration is grounded in a number of principles of correct reasoning, some of these principles – the fundamental or basic laws of logic, which Russell refers to as the indemonstrables – must themselves be incapable of demonstration: that is, they cannot be given deductive justifications in the sense of a demonstration of their correctness. But there is more to the passage than this observation, and this more has to do with Russell’s comments on the possibility of independence proofs, when these are applied to the ‘principles’ or ‘axioms’ of a logical axiom system. The model for such proofs that Rus- sell is relying here is familiar from other contexts. The basic idea is as follows. An axiom ơn is shown to be independent of other axioms, ơ1...ơn-1, by finding an interpretation that makes ơ1...ơn-1 true and ơn false. If such an interpretation exists, that shows that ơn is not a logical consequence of the rest of the axioms, and is in that sense independent of them. Independence proofs are just one application of the general idea behind semantic reasoning, to wit, that the logical properties of a sentence are determined by its behaviour with respect to truth-values across relevant models or interpretations. Thus, Russell’s views on independence proofs can be used to throw light on the more general question regarding his attitude towards the notion of Ch. 4 Logic as the Universal Science I 419 alternative interpretations. Let us begin by noting that Russell’s description of independence proofs is ambiguous in much the same way that modern characterizations of model-theoretic reasoning are. Consider a typical description of independence proofs: “To show that an axiom ơ is independent, give a model in which all axioms but ơ are true, the inference rules are sound, but ơ is false” (Zach 1998, p. 348). Here the instruction to find a model can be understood in either of the following two ways. Firstly, it can be taken to embody a counterfactual strategy; this amounts to considering what would happen to the truth-values of statements in certain counterfactual situations. Secondly, “giving a model” can refer to a reinterpretation strategy, whereby one reinterprets certain of the expressions occurring in the relevant statements and considers the truth-values of the resulting statements. A similar ambiguity is also present in Russell’s text. Which one of the two strategies is relevant to the passage depends upon the sense attached to the expressions “axiom” and “supposing an axiom false”. Does “supposing an axiom false” mean imagining a counterfactual situation in which an axiom is false? Or does it mean reinterpreting one or more of the expressions occurring in axiom, or assigning an interpretation to a sentence-schema, in such a way that the resulting axiom-sentence is false? On the face of it, Russell’s non-linguistic conception of propositions commits him to the counterfactual strategy. Since propositions possess determinate truth-values in their own right, there does not seem to be any room, in Russell’s logic, for sentences, or sentence- schemas awaiting interpretation. Hence, it seems, ‘supposing an axiom false’ can only mean imagining a counterfactual situation in which a proposition is false. However, what Russell has to say about independence elsewhere in the Principles provides conclusive support to the alternative interpretation. In §120, where Peano’s axioms for arithmetic are introduced, Russell mentions that Peano’s exposition of the subject enables one to prove, among other things, that if the primitives of the 420 Ch. 4 Logic as the Universal Science I axiomatisation are “regarded as determined by [Peano’s five axioms]”, then the axioms are mutually independent. And he explains further that this is done by finding for each set of four axioms an interpretation which renders the remaining axiom false. In §121 actual instances of this procedure are given. For instance, “[g]iving the usual meanings to 0 and successor, but denoting by number finite integers other than 0”, all Peano’s axioms come out true, except the first one, which states that “0 is a number”. Evidently, what is discussed here is the reinterpretation strategy; at least in these two sections, then, Rus- sell regards “propositions” as sentences capable of a multiplicity of meanings; thus, also, axioms are subject to varying interpretations.153 Even more revealing is the following quotation from §8 of the Principles:

153 Furthermore, if “supposing an axiom false” is construed as in the counterfactual strategy, Russell would have had a very simple argument against the method of interpretation. In §430 of the Principles we find the following remark: “there seems to be no true proposition of which there is any sense in saying that it might have been false. One might as well say that redness might have been a taste and not a colour. What is true, is true; what is false, is false; and concerning fundamentals, there is nothing more to be said.” It follows immediately from a view like this that the counterfactual strategy must be written off as illegitimate; propositions are simply true and false, according to Russell, and hence it does not even make sense to consider the possibility that under different circumstances a proposition that is true might have been false. Russell’s view is likely to strike us as odd, but it unquestionably was his view. But he does not use this view as a general argument against the notion of alternative interpretations; he does not claim, that is, that independence proofs are impossible, because they involve an impossible procedure, namely one that is based on the assumption that a proposition that is in fact true could have been false. In §17 he is clearly not endorsing any such charge of impossibility or not making sense. Admittedly, this observation is less than conclusive, since we cannot assume without further ado that a claim Russell makes in one section of the Principles carries automatically over to another. But it must be allowed some weight. Ch. 4 Logic as the Universal Science I 421

So long as any term in our proposition can be turned into a variable, our proposition can be generalized; and so long as this is possible, it is the business of mathematics to do it. If there are several chains of deduction which differ only as to the meaning of the symbols, so that propositions symbolically identical become capable of several interpretations, the proper course, mathematically, is to form the class of meanings which may attach to the symbols, and to assert that the formula in question follows from the hypothesis that the symbols belong to the class in question. In this way, symbols which stood for constants become transformed into variables, and new constants are substituted, consisting of classes to which the old constants belong. Cases of such generalization are so frequent that many will occur at once to every mathematician, and innumerable instances will be given in the present work. Whenever two sets of terms have mutual relations of the same type, the same form of deduction will apply to both. For example, the mutual relations of points in a Euclidean plane are of the same type as those of the complex numbers; hence plane geometry, considered as a branch of pure mathematics, ought not to decide whether its variables are points or complex numbers or some other set of entities having the same type of mutual relations.

Although Russell’s wording is not that of a present-day mathematician’s or logician’s, it is nevertheless remarkably modern. Passages like these make it quite evident that Russell has no objections of principle to alternative interpretations; on the contrary, as the quotation from §8 shows, he endorses, rather than rejects, the idea that symbols are capable of a multiplicity of meanings or interpretations. Of course, the fact that in passages like these Russell speaks about symbols rather than what they stand for, suggests that his commitment to ontologically construed propositions as the true subject- matter of logic is not quite as simple and straightforward as our previous discussion indicates – and, we may add, as is generally assumed. On the one hand, his official view of the matter is that logic is concerned with propositions and these are non-linguistic entities (what sentences mean, rather than the sentences themselves, we should have to say, if we want to preserve the connection with language). In this official sense propositions are simply not entities to which semantic considerations in the customary sense would be rele- 422 Ch. 4 Logic as the Universal Science I vant.154 This applies even to denoting concepts; these are symbolic entities, but this does not mean that the denotation of a denoting concept is in any sense up to us, or that a denoting concept does not have a denotation until it is assigned one.155 On the other hand, at least on occasion he treats propositions themselves rather like sentences, or else conducts his discussion of logical matters in linguistic terms.156 It would be going too far to suggest that Russell’s notion of proposition is actually ambiguous between a non-linguistic and linguistic sense. The moment he begins to “philosophize” about propositions, it becomes at once evident that propositions are worldly entities, rather than linguistic or mental representations.157 It is more plausible to argue that what we have here is yet another instance of the tension between Peanist and Moorean elements in Russell’s logic.

154 But see below, section 4.5.5, where it is pointed out that a notion of interpretation can be applied even to Russellian propositions. 155 See Principles, §56. 156 This is not restricted to the few short discussions of interpretation that we have cited, but occurs quite extensively in the Principles, for example, in Russell’s discussion denoting. Although, officially, denoting has nothing whatsoever to do with linguistic expression, Russell in fact constantly formulates his point in linguistic terms: see §§56, 58 and 65. The fact that Russell so effortlessly moves back and forth between use and mention can be explained, at least in part, by the close affinity between sentences and propositions: at this time he accepted the idea that propositions are (almost exactly) isomorphic to the sentences which express them. Thus, in Russell’s proto- semantics propositions are regarded as sentential meanings; hence, apparently, it makes little difference whether one discusses sentences or propositions (similarly for their constituents). It should be kept in mind, however, that when an author like Russell uses apparently linguistic terminology, this does not necessarily imply that what is being discussed is something linguistic. Nicholas Griffin reminded me that this way of using apparently linguistic terminology is common in Bradley, who uses ‘adjectives’ for properties, for instance (we have, indeed, seen one instance of this usage above, in section 4.4.2), and is also adopted by Moore (Moore (1899) is a good example). 157 See, for instance, the beginning of (Russell 1904a) as well as (Russell 1905c). Ch. 4 Logic as the Universal Science I 423

Very briefly, the Moorean notion of proposition appears to be in some sense oblivious to such novelties as the idea of alternative interpretations. This is hardly surprising, since we may safely assume that Moore knew nothing about them at the time he introduced the notion. Peano’s logic, by contrast, can readily accommodate such ideas through the notion of propositional function.158 I conclude that Russell’s conception of logic in no way excludes the notion of alternative interpretations (as we shall see in a moment, this statement requires one modification). What then, is the argument that Russell advances in §17 of the Principles? What is found there is not a general argument against “alternative interpretations”, but a more specific one with a strictly limited scope; the point of the passage is that a method which has been found applicable elsewhere – find an interpretation which makes one axiom false and the rest of the axioms true – cannot be applied to the particular case of a logical axiom-system. The reason is that the axioms of a logical axiom- system enjoy a special status with respect to our reasoning, including reasoning about independence and other such properties of a logical calculus. In other words, Russell’s argument turns on the fundamental character of these axioms; since they model the principles of correct reasoning, reasoning itself becomes impossible, if they are denied. That this is Russell’s argument is confirmed by his letter to Philip Jourdain, 28 April, 1909. In an earlier letter, Jourdain had posed him a question about the independence of logical axioms:

When you enumerate the primitive propositions of logic, do you prove their independence by the usual method of giving certain interpretations to the primitive ideas, so that all but one (in turn) of the primitive propositions is verified? (Grattan-Guinness 1977, p. 117)

Note that by the “usual method” Jourdain clearly means what we called the reinterpretation strategy. Russell gave the following reply to

158 For further discussion, see section 4.5.5. 424 Ch. 4 Logic as the Universal Science I

Jourdain’s query:

I do not prove the independence of primitive propositions in logic by the recognised methods; this is impossible as regards principles of inference, because you can’t tell what follows from what supposing them false: if they are true, they must be used in deducing consequences from the hypothesis that they are false, and altogether they are too fundamental to be treated by the recognized methods. (ibid.)

Russell’s objection to the use of ‘recognised methods’ is not that reinterpretation cannot be applied to the entities with which logic is concerned; what he argues is that the principles of inference are too fundamental to be treated by the reinterpretation strategy. In order to reason about independence, we must use principles of inference, and therefore we cannot assume, even hypothetically, that such principles are false. The reasoning here is similar to the argument about the non- justifiability of the fundamental laws of logic that we discussed earlier. These laws are presupposed, whenever we reason about some subject-matter; they must be used in deducing consequences, as Russell puts it in the letter to Jourdain. Therefore these laws themselves cannot be justified inferentially. And if they are not true – or valid – the consequences they seem to license do not really follow, as Russell points out in §17 of the Principles. Therefore these laws cannot be assumed false for some specific purpose, either. There is a further similarity between the two cases. Russell’s point about independence proofs seems sound enough, as long as it is applied to the “principles of correct reasoning”; as is mentioned in Zach’s description of the content of independence proofs, such proofs presuppose that the inference rules are sound. Hence the rules cannot be assumed false, which is precisely what Russell argues. This conclusion can be circumvented, however, if we redirect the discussion in the same way as we did when we considered the question of justification. That is, if the focus is shifted from the laws of logic to their symbolic or linguistic expression, such proofs (soundness and inde- Ch. 4 Logic as the Universal Science I 425 pendence) can be given. From this point of view Russell is simply mistaken about independence proofs. When a logical axiom system is tested for independence, its axioms and rules of inference are not treated as principles of correct reasoning; they are treated without reference to any specific interpretation that can be assigned to them.159 Independence proofs in this sense concern certain syntactically specified objects (string of symbols) generated by the syntactic rules of the formal language whose formulas are under scrutiny. If, on the other hand, one insists that the true subject-matter of logic are the principles themselves, rather than the language or symbolism in which such principles are formulated, one may well be inclined to think that such proofs are not very important. Had Russell considered independence proofs from this perspective, he might well have drawn just this conclusion, given his prevalently non-linguistic conception of logic. There is, indeed, evidence that he did consider “interpretation” from precisely this point of view. In *5 of The Theory of Implication Russell seems to be making just this point. The passage is worth quoting at length:

Treated as a “calculus”, the rules of deduction are capable of many other interpretations. But all other interpretations depend upon the one here considered, since in all of them we deduce consequences from our rules, and thus presuppose the theory of deduction. One very simple interpretation of the “calculus” is as follows: The entities considered are to be numbers which are all either 0 or 1; “p q” is to have the value of 0 if p is 1 and q is 0; otherwise it is to have the value 1; ~p is to be 1 if p is 0, and 0 if p is 1; p . q is to be 1 if p and q are both 1, and is to be 0 in any other case; p q is to be 0 if p and q are both 0, and is to be 1 in any other case; and the assertion-sign is to mean that what follows has the value 1. [...] Symbolic logic considered as a calculus has undoubtedly much interest on its own account; but in my opinion this aspect has hitherto been too much emphasized, at the expense of the aspect in

159 Cf. Kneale and Kneale (1962, p. 690). 426 Ch. 4 Logic as the Universal Science I

which symbolic logic is merely the most elementary part of mathematics, and the logical prerequisite of all the rest. (Russell 1906d, pp. 183-184)160

Given a passage like this, there can really be no doubt that Russell had no objections of principle to make against “interpretation”; when the rules of deduction are considered as a calculus, they can be interpreted in different ways; whatever interest these different interpretations may have, though, they are, all of them, dependent upon the one that regards them as principles of correct reasoning. Thus we reach the following conditional conclusion: insofar as Russell associates the subject-matter of logic with propositions in the ontological sense – sense that is connected with taking the subject- matter as constituting the principles of correct inference – this gives him a reason to ignore independence proofs for a logical axiom system. §17 of the Principles does not license any stronger conclusion than this, however; in particular, it does not warrant the claim that Russell’s conception of logic would somehow preclude or be anti- thetical to the notion alternative interpretations per se. Indeed, as we have just seen, there is very clear textual evidence for the contrary conclusion, or passages where he clearly states that alternative interpretations are perfectly legitimate. On the face of it, insofar as we accept the relevant passages in the Principles as indicative of Russell’s considered views on the matter, this commits us to the view that there is, on his part, same wavering between a linguistic and non-linguistic sense of “proposition” or the subject-matter of logic. Russell’s official notion of proposition is one that clearly prevents us from applying such usual terminology as “interpretation” or “giving meaning” to propositions. That he nevertheless uses this terminology (as in sections 120-122 of the Principles) is presumably to be explained by its naturalness; or by the fact that even though sentences and other linguistic entities are officially of no concern to logic, according to Russell, there is none the less an extremely

160 This passage is repeated, verbatim, in Principia; see Whitehead and Rus- sell (1910, p. 115). Ch. 4 Logic as the Universal Science I 427 close connection between sentences and their parts, on the one hand, and propositions, on the other. This connection is grounded in the assumption, which he never explains but simply assumes to be valid, that there is a rough one-one correspondence between sentences and propositions, so that the latter are capable of acting as sentential meanings, in addition to their other roles. Most likely, though, is the explanation that Russell simply paid little attention to such subtleties of logical theory as a strict delineation between sentences and propositions.

4.5.5 “Interpretation” and Semantics

In section 4.5.3 it was argued that Russell’s anti-semanticism should be seen as a consequence of the fact that he did not, at this point, have room for the concept of truth-condition. Now, it is natural to think that use of a concept like “interpretation” presupposes a linguistic perspective on logic. Moment’s reflection shows, however, that this is incorrect. For, even to someone who does not think that logic would be a matter of language, there is still available a notion that is at least analogous to “interpretation”. Re-interpretation is naturally regarded as a procedure which, when applied to some symbols, allows a systematic change in the objects their stand for.161 A simple example shows, however, that the idea of

161 These symbols may be thought of in two different ways. Firstly, it may be assumed that they are symbols which do not initially have any meaning (reference) at all, but receive one through interpretation; since this interpretation is up to us, the assignment of meanings (references) can be done in different ways, i.e., interpretation can be varied at will. The only restriction on interpretation and re-interpretation is the principle that semantics must track syntax: syntax determines what kinds of entities may be assigned to which symbols (individual constants to members of the domain, n-tuples of individuals to n-place predicates, etc.). Secondly, symbols may be regarded as entities that possess initial meanings and references, but these are nevertheless subject to systematic re-interpretation (within limits, as in the first case). 428 Ch. 4 Logic as the Universal Science I re-interpretation itself is much older than any explicit or systematic semantics based on the notion variation of interpretation. Consider an argument like

(A) Every dog is a mammal Some mammals have tails Some dogs have tail

(A) has true premises and a true conclusion. But the argument is clearly not valid. As logicians have been pointing out ever since Aris- totle, this can be shown by finding an argument which is “of the same form” as (A) but whose premises are true and conclusion false. However, the relationship between (A) and this second argument admits more than one description. For example, we can explain that a counterexample to the validity of (A) emerges from letting “dog”, “mammal” and “have tail” change their meanings and reference; more simply, since we are interested only in the determination of truth-values, we may bypass the level of meaning, and speak directly of assignments of references (semantic values) to the relevant parts of the sentences featuring in (A); thus, we can let “dog” refer to human beings, “mammal” to animals in general and “have tails” to things that live under the sea. Reinterpreting parts of sentences in this way yields an argument that has true premises and a false conclusion, which shows that (A) is not valid. But this is not the only way to describe what we do when we demonstrate the invalidity of an argument. We might have said as well: substitute (the expression) “human being” for “dog”, “animal” for “mammal” and “live under the sea” for “have tail”. This procedure of substituting expressions for expressions results in the following ar-

Obvious though this distinction is, pointing it out at least serves to remind us of one important thing; the fact that one regards logic as a full-fledged language or as a system of symbols that possess meanings and references, in no way dictates one’s attitude towards the possibility of reinterpretation.; cf. here Tappenden (1997), pp. 240-3. Ch. 4 Logic as the Universal Science I 429 gument:

(B) Every human being is an animal. Some animals live under the sea. Some human beings live under the sea.

Since (B) has true premises and a false conclusion, we have shown that (A), an argument that is “of the same form” as (B), is invalid. It would seem that there is but a small step from this latter procedure, whereby expressions were substituted for expressions to give an argument that is of the same form as the original argument but be- haves differently with respect to truth-values, to the procedure that Russell described in the Principles. Since logic is not a matter of language for him, he does not speak of sentences and expressions (most of the time, anyway); rather, as we have seen, he uses the terminology of propositions and their constituents. The linguistic version of the notion of reinterpretation, however, is easily translated into the language of propositions; the basic idea that underlies both accounts is appropriately dubbed the substitutional account of logical form. Evidently, insofar as one assumes a Russellian notion of proposition, one should not, strictly speaking, say that a term – an entity like Socrates or the property of wisdom – is substituted for a constituent of a proposition, or that a term occurring in a proposition “gets replaced” by another term (as Russell occasionally does; see, e.g., §22 of the Principles). To that extent the term “substitutional account”’ is a misnomer; propositions in Russell’s sense and their constituents are simply not the sort of entities that can replace and get replaced by other entities. But talk of “substitution” is conveniently short and is unlikely to give rise to any genuine confusion or misunderstanding. For the sake of perspicuity, we should nevertheless say, not that a term in a proposition gets replaced by another term, but something like “given a proposition, consider another proposition that is exactly like the original one, except for containing ... as a constituent, where the original propositions contains --- as a constituent”. Understood in 430 Ch. 4 Logic as the Universal Science I this way, the framework of Russellian propositions readily yields a non-linguistic variant of the substitutional account of form. The counterexample strategy, for example, is readily available to an advocate of Russellian propositions. In order to demonstrate the invalidity of an argument like (A), we consider another argument, like (B), whose premises and conclusion – construed now as Russellian propositions – stand in the “same form”-relation, as Russell con- ceives of it; hence, the fact that there is an argument like (B) shows that (A) is invalid. These considerations suggest that there is nothing specifically semantic about the idea of variation, and that the connection emerges only when this idea is paired with the further notion of interpretation. The notion of (semantic) interpretation should therefore be seen as a refinement of a more primitive, partly intuitive notion of variation, which is available to a logician quite independently of his attitude towards “semantics” in something like its present-day, technical sense.162 This observation goes directly against the view, sometimes found in the secondary literature, that the difference between the universalist and calculus conceptions can be formulated by referring to the question of whether variation or “reinterpretation” is possible. For instance, Kusch (1988, pp. 12-13) argues that the basic idea of model- theory is to be found in the idea that, given a set of symbols and strings of symbols generated by syntax, one may interpret them in different ways, which means: let them refer to different objects. He argues, furthermore, that this idea is unavailable to the universalist logician, who thinks that the relations holding between language and reality have been fixed, or given once and for all, and cannot for that reason be changed. Supposing, for the sake of argument, that this is what the early Russell thought about semantic relations,163 the fact remains that variation can still be retained, if one considers, not se-

162 Cf. here Dummett (1993, pp. 23-24). 163 And since he was silent on the issue, we cannot really decide what views, if any, he held on the “possibility” of semantics. Ch. 4 Logic as the Universal Science I 431 mantic relations, but simply expression classes or what these expressions stand for (even if in this latter case the terminology of “variation” becomes somewhat figurative). Given this conclusion, we must conclude that there is no essential connection between semantics and the notion of variation. On the contrary, the notion of variation is one attempt to get hold of the notion of schematic representation, or form, if we use classical philosophical idiom; as such, it is independent of the idea of semantics, although one can, of course, introduce it into semantics as well, as is shown by the specifically semantic notion schematic representation qua (semantic) interpretation; the very idea of logic presupposes some notion of form or schematic representation, and the notion of variation is available for that purpose, quite independently of one’s attitude towards semantics and its possibility. Several conclusions follow. Firstly, the notion of interpretation can be given more than one sense. Even if we found a Russellian argument to the effect that some notion of “interpretation” is illegitimate, it would not follow that he was opposed to the general idea involved in the notion of alternative interpretations, or that he was opposed to each and every use of interpretation.164 Secondly, once we realise that “interpretation” is susceptible to a number of readings, we can also see that there need not be much of a connection between that notion and semantics in something like the modern sense of that term. If there is one reason why the early Rus- sell shunned semantics, that reason is to be found in the view, clearly expressed, if not quite consistently followed up in the Principles, that linguistic meaning is irrelevant to logic. This reason, however, has very little to do with his universalism about logic. But we should also observe that talk of Russell as someone who “shunned” semantics is itself problematic. And it is problematic precisely because it suggests that the early Russell’s perspective on logic shares an important presupposition with ours. Had he been exposed

164 Furthermore, our discussion of the early Russell has not revealed any such argument by him. 432 Ch. 4 Logic as the Universal Science I to Tarskian semantics, he might well have failed to appreciate its import. It is not clear, however, what interest this counterfactual has and what insight it provides into his conception of logic. Indeed, it seems to be no more than a needlessly complicated way of stating the obvious, namely that the early Russell’s reflections on the nature of logic predate the linguistic turn in philosophy; when we consider Russell’s conception of logic, there really is no need to back up the anti-semantic conclusion by complicated considerations concerning content and its fixity. On the contrary, the conclusion follows as a simple corollary from his predominantly non-linguistic conception of the subject-matter of logic. In other words, when it comes to Rus- sell’s conception of logic, we would do well to keep questions about the notion of interpretation separate from questions about semantics. Thirdly, following this instruction, we see that there is little reason to attribute the fixed-content argument ((FCA)) to Russell, as this argument was formulated above in section 4.5.2.3.1; (FCA) presupposes a framework, namely a linguistic conception of logic, that was, for the most part, foreign to the early Russell; its conclusion speaks of “semantic explanation”’, by which is meant an argument to the effect that a sentence possesses this or that semantic property, given some particular interpretation of its constituent expressions. Such considerations, however, play a minor role in the early Russell’s logic – not because they are illegitimate (that they clearly are not, according to Russell, if also not very important), but for reasons we have just explained. Fourthly, the question where the early Russell and his conception of logic stand vis-à-vis model-theoretic reasoning should be decided not by considering his views on “semantics” but by determining how his version of the substitutional account of form differs from the gist of model-theoretic reasoning. Considering this question, it must be admitted that there are genuine and non-trivial differences between Russell and model-theory. This will be the subject of the next subsec- tion. Ch. 4 Logic as the Universal Science I 433

4.5.6 Generality and Quantification

4.5.6.1 Unrestricted Generality

So far our discussion of interpretation has revolved round the notion of variation. There is another dimension to interpretation, however: if our language permits the expression of generality – and we surely want our language to be rich enough to permit us to talk about what is, and is not, generally the case – we must indicate the range of our generalizations. In model theory this is effected by including in a model a domain or “universe of discourse”. This consists in a set of entities with respect to which the formulas of the language are interpreted. Hence, even if quantifiers appear to speak of everything and something simpliciter, they are in fact restricted to the members of the relevant domain. The interpretation of formulas, when it is done as in model theory, thus appears as a clear example of restricted quantification, with the domain-of-discourse providing the relevant restriction. Russell’s understanding of quantification is conspicuously different from this; like Frege, he does not think of quantification as in this way restricted. Rather, he regards universal and existential quantifiers as logical constants indicating, respectively, unrestrictedly or absolutely everything and non-emptiness. As we have seen (cf. section 4.4.10.1), Rus- sell’s technical treatment of quantification in the Principles follows Peano: occurs only as subscripted to conditionals, yielding the notion of formal implication, and only as attached to class-concepts (thus the existential “quantifier” is not a variable-binding operator at all). These eccentricities aside, the important thing about Russellian quantification is precisely the idea that quantifiers are logical constants; in accordance with Russell’s concept of logical constant, this means first and foremost that quantifiers always come with a wholly unrestricted domain: it would be an entirely inappropriate deviation from the universality of logic to regard its application in any way restricted.165 When we wish to impose such restrictions – when, for

165 This explanation is one that has not been yet discussed; see below, 434 Ch. 4 Logic as the Universal Science I example, we are not interested in everything simpliciter but every thing of such-and-such kind – they must be understood against the background of unrestricted generality. This point is made explicitly in §7 of the Principles:

It is customary in mathematics to regard our variables as restricted to certain classes: in Arithmetic, for example, they are supposed to stand for numbers. But this only means that if they stand for numbers, they satisfy some formula, i.e. the hypothesis that they are numbers implies the formula. This, then, is really what is asserted, and in this proposition it is no longer necessary that our variables should be numbers: the implication holds equally when they are not so. Thus, for example, the proposition “x and y are numbers implies (x + y)2 = x2 + 2xy + y2” holds equally if for x and y we substitute Socrates and Plato: both hypothesis and consequent, in this case, will be false, but the implication will be true. Thus in every proposition of pure mathematics, when fully stated, the variables have an absolutely unrestricted field: any conceivable entity may be substituted for any one of our variables without impairing the truth of our proposition.166 section 5.4. 166 Of course, formal implication and are not Russell’s only explanation of quantification. As we have seen, he entertains the hope that conceptual analysis would reveal denoting concepts behind the formal apparatus of quantifiers and variables. Speaking quite generally, the notion of unrestricted variable is associated in Russell’s thought with the Peanist treatment of quantification, whereas the theory of denoting concepts typically operates with restricted variables. A denoting concept /Dơ/ denotes, roughly, a special plural object: what objects are denoted is determined by the class- concept ơ, and what sort of object is denoted is determined by the concept /D/. In modern terminology, such quantifiers are called restricted quantifiers (“all numbers”, “some apostles”, “any cat”). They do not range over everything, but are restricted to the members of the class indicated by the class-concept. However, even here one can easily introduce unrestricted generality by choosing a sufficiently abstract or general noun, like ‘thing’ or ‘entity’ or Russell’s ‘term’. As we have seen, Russell’s preferred analysis of quantification in the Principles was that the denoting concept /any term/ is, or denotes, the true variable, the characteristic mark of which is precisely that its range is unrestricted. This means that the plural object it denotes Ch. 4 Logic as the Universal Science I 435

In this passage Russell not only argues that, as a matter of fact, the variables in propositions of pure mathematics or logic range over everything simpliciter. He also argues against the idea of irreducibly restricted quantification; in his view unrestricted quantification is more fundamental than restricted quantification, and therefore the latter presupposes the former. His argument for this conclusion is simply that if we are to restrict the domain or universe of discourse in any way, the restriction must be indicated by a suitable explicit statement. But the statement expressing the restriction is itself a quantified statement, and in this second statement variables must be understood as ranging unrestrictedly over everything. More precisely, the claim is that restricted quantification can be incorporated by introducing a suitable predicate which applies to all and only the members of the restricted domain. In this way, Russell explains, “(x + y)2 = x2 + 2xy + y2”, in which the variables appear to be restricted to numbers, gives way to “if x and y are numbers, then (x + y)2 = x2 + 2xy + y2”, a statement in which ‘x’ and ‘y’ “have an absolutely unrestricted field”. In the light of this argument, then, unrestricted quantification appears as the more basic.167

4.5.6.2 Hylton on Russell on Generality

According to Hylton (1990a, p. 202, 1990b, pp. 206-7), Russell’s argument for the conceptual priority of unrestricted quantification highlights an essential aspect of his universalist conception of logic. includes absolutely every term. And since terms constitute the all-inclusive category, it follows that everything (that is, everything there is) is included in the denotation of /any term/. 167 This argument is repeated in Russell (1906a, p. 205) and Russell (1908, p. 71). In the former text, Russell mentions the notion of a “universe of discourse”, introduced by “the older symbolic logicians”. If valid, Rus- sell’s argument undermines this notion in the sense of showing that understanding “universe of discourse” presupposes grasp of the more primitive notion of “unrestrictedly all”. 436 Ch. 4 Logic as the Universal Science I

Here is how Hylton explains the presuppositions of the argument:

This argument of Russell’s takes it for granted that the statement which establishes the universe of discourse is on the same level as the assertion which is made once the universe of discourse is established. Thus the former can be taken as antecedent and the latter as consequent in a single conditional statement. Russell, that is, assumes that all statements are on the same level; this contrasts with the model-theoretic view that we must distinguish some as object-language statements and some as metalanguage statements. Intrinsic to Russell’s conception of the universality of logic is the denial of the metalinguistic perspective which is essential to the model-theoretic conception of logic. (1990b, p. 207)

According to Hylton, the doctrine of unrestricted variables is an integral part of Russell’s universalist conception of logic. He argues, furthermore, that this doctrine is fundamentally different from what is found in model-theory. The point is this. Consider a pair of statements like

(1) [ƶx, and

(2) let the variables of the object language range over natural numbers.

In model-theory, the relationship between (1) and (2) is considered somewhat as follows. Sentence (1), which asserts that a certain condition holds of “all entities”, belongs to an object-language, L. In fact, however, for (1) to make any assertion at all, one needs an explicit stipulation, belonging to a metalanguage, L*, which specifies a family of interpretations or models for the sentences of L, including (1), and this includes a specification of a universe-of-discourse for the variables of L to range over. In other words, since the strings of symbols that are the object language sentences express content only after they have been given interpretations, the variables occurring in (1), con- Ch. 4 Logic as the Universal Science I 437 sidered on their own, are silent; what they range over must be stipulated separately, and this stipulation is expressed in a separate sentence belonging to of L*. Russell, by contrast, thinks that “all statements are on the same level”. It is for this reason, according to Hyl- ton, that Russell, so to speak, runs object-language and metalanguage together, thereby combining (1) and (2) into a single conditional statement:

(3) x(if Nx, then ƶx), where “Nx” is a predicate expressing to what kind of entities the formula “ƶx” applies. One might think that the model-theoretician’s assumption of a hierarchy of languages offers a straightforward argument against the absolutist conception of generality: as soon as one distinguishes between object- and metalanguages, one can stick to a conception of generality that is irreducibly relativistic.168 Whether or not this is Hyl- ton’s view is not clear. It may be that he is merely pointing out what he takes to be an important difference between Russellian and model- theoretic treatments of quantification, without drawing any further consequence from that difference. Be that as it may, one could argue that a proponent of the relativist conception can always resort to a distinction between languages, contexts or different kinds of acts in arguing for his preferred treatment of generality. It is not clear, however, that Russell’s argument can be so lightly dismissed. For one could defend the conclusion, if not his particular argument for that conclusion, as follows. Understanding the metalan-

168 Insofar as we address the problem more generally, and not merely from the standpoint of the contrast between model-theory and Russell’s views, we could speak of different contexts, rather than language – this would be a natural choice in view of the fact that restrictions on quantification are a special-case of context dependence – or even acts, pointing out a difference between an act of assertion, and an act of explicitly stipulating a range for quantifiers. 438 Ch. 4 Logic as the Universal Science I guage stipulation (2) is a necessary condition for grasping the intended content of the object-language assertion (1). Hence, there is a connection between (1) and (2), even though they do not belong to one and the same language. And if it is correct that unrestricted quantification or absolute generality is the norm, this still holds, even if the expression of the norm and the assertion to which the norm applies are separated into distinct languages. Of course, the primacy of absolute generality needs to be argued for. One suggestion, following Williamson (2003, sec. VI), would be as follows. The argument starts from what Williamson calls kind- generalizations. These are assertion like “all penguins waddle” which generalize over all members whatsoever of the relevant kind. As Wil- liamson points out, there is no obvious reason why we should have to introduce absolutely unrestricted quantification in handling such generalizations; quantifying over members of the relevant kind does not seem to involve quantifying over absolutely everything. Closer inspection suggests, however, that a generality-relativist will in fact face considerable difficulties in trying to incorporate kind-generalizations into the framework of relative generality. Consider the kind-generalization “all penguins waddle”. For an utterance of this sentence to express the information that absolutely all penguins waddle – and it seems clear enough that we do express such generalizations, both in scientific and everyday contexts – it is not enough if the relevant domain does as a matter of fact contain absolutely all the penguins there are (id., p. 437); rather, one must already have access to the information that the relevant domain includes absolutely every penguin; if I erroneously believe that there are penguins in the Arctic that do not waddle, my utterance of “all Ant- arctic penguins waddle” will not express the information that absolutely all penguins waddle, even if there are no penguins except in the Antarctic. If one suggests that one can simply resort to stipulation here, the reply can be made that the stipulation is something that needs to be said (cf. the relationship between sentences (1) and (2) above). As Williamson (id., pp. 437-8) points out, the crucial point is Ch. 4 Logic as the Universal Science I 439 that the relativist must find a way of expressing the information that absolutely every thing of kind K is in the domain over which one is quantifying, and now the question arises: how can this information be expressed without resorting to quantification over absolutely everything? It seems we are here close to the point that Russell made in the Principles, §7. This question is particularly acute in the case of model-theory. For in standard model-theory, meta-language statements are needed to assign semantic values (interpretations) to syntactically specified formulas of the object-language, and this includes an explicit statement, or stipulation, to the effect that object-language quantifiers are to range over such-and-such entities. There are various ways in which the generality-relativist may try to resist these lines of thought. These will not be considered here.169 It nevertheless appears that the conclusion of Russell’s argument cannot be so easily dismissed, even if the form in which he presents it is a little too simple.

4.5.6.3 Criticism of Hylton’s Reading

Another question relating to Russell’s argument, and more topical in the present context, is this: What were Russell’s reasons for maintaining the doctrine of unrestricted variables? As we have seen, Hylton argues that the underlying reason is the view that “all statements are on the same level”. In other words, the argument spells out one consequence of the denial of the meta-linguistic perspective, which Hyl- ton takes to be intrinsic to Russell’s universalist conception of logic. I do not think this explanation given by Hylton is the correct one; since Russell does not – most of the time, anyway – think about logic in linguistic terms, the denial of a metalinguistic perspective can hardly be intrinsic to his thinking about logic. And if we take this into account, substituting something else for “linguistic”, the point remains that Russell’s conception of logic is perfectly compatible with a meta-

169 Again, see Williamson (2003). 440 Ch. 4 Logic as the Universal Science I perspective on logic. The real reason why Russell held the doctrine of unrestricted variables is to be found, again, in his notion of proposition. The view that I am attributing to Russell is as follows. As we have seen, a Russellian proposition is in no way a linguistic – more generally, representational – entity. This means, among other things, that the possessing of a particular truth-value is an intrinsic feature of a proposition. Hence, anything that is relevant to the determination of a proposition’s truth-value must be given as soon as the proposition is given. It need not be literally a part or constituent of the proposition, although this is how Russell tends to think about the issue. Ap- plied to quantified propositions, this means that the range of quantification must be somehow internal to the proposition. Insofar as quantification is construed in terms of denoting concepts, this feature is secured by Russell’s view that a denoting concept “inherently and logically” denotes the terms that constitute its denotation (Principles, §56). Calling this relation “logical” is presumably meant to exclude any linguistic, psychological or otherwise conventional element from the fundamental relation of denotation that holds between certain kinds of concepts and terms. Russell makes precisely this point in the following quotation from The Theory of Implication, a passage which contains a more elaborate version of the argument first given in the Principles, §7:

The possible values of an independent variable are always to include all entities absolutely. The reason for this is as follows: If we affirm some statement about x, where x is restricted by some condition, we must mention the condition that make our statement accurate; but then we are really affirming that the truth of the condition implies the truth of our original statement about x; and this, in virtue of our interpretation of implication, will hold equally when the condition is not fulfilled. The “universe of discourse”, as it has been called, must be replaced by a general hypothesis concerning the variable, and then our formulae are true whether the hypothesis is verified or not, because an implication holds whenever its hypothesis is not true. The old theory of the “universe” had the defect of introducing tacit hypotheses, thus making all enunciations incomplete, since a hypothesis does not cease to be an essential part of a proposi- Ch. 4 Logic as the Universal Science I 441

tion merely because we do not take the trouble to state it. (Russell 1906d, * 1.3 (pp. 162-163; italics added)

Russell’s claim that “a hypothesis does not cease to be an essential part of a proposition merely because we do not take the trouble to state it” contains precisely the view that was attributed to him above; a proposition being what it is, everything that contributes to the determination of its truth-value must be essential to it (either intrinsic or internally related to it), for otherwise the truth-value of a proposition would dependent upon something else. It is this kind of reasoning, rather than a denial of the possibility of metaperspective, that underlies Russell’s views on quantification.

4.5.7 Russell’s Concept of Truth

4.5.7.1 Preliminary Remarks

The above quotation from the The Theory of Implication is very useful, as it brings out very clearly how Russell’s conception of proposition implies a rejection of a semantic perspective on logic, understood after the manner of model-theory. Thus, in many cases it can be plausibly argued that his way of looking at logical theory is determined by a non-linguistic conception of the subject-matter of logic, that is, by the notion of Russellian proposition. Here, then, we have an apparently compelling and very straightforward explanation of why he largely ignores a semantic account of logic. Of course, as we have also seen, this conclusion must be qualified in the sense that mere reference to a rejection of a semantic perspective cannot as such explain very much: correctly understood, it does not imply the impossibility of metaperspective, or of a notion of interpretation, etc. This, however, is not the only respect in which the explanation needs a qualification. Let us give the name “strictly semantical” to any account of logic that presupposes the linguistic approach. Clearly, Russell’s notion of proposition, if followed out strictly, excludes this 442 Ch. 4 Logic as the Universal Science I approach. On the other hand, there are at least some concepts which are prima facie semantic but can nevertheless be applied to Russellian propositions. Most importantly, Russell thinks that propositions in his sense are the ultimate truth-bearers. They are also entities that possess logical properties and stand in logical relations to one another. It would seem, then, that he has good reasons for using the concept of truth in logical theory, quite independently of what he may have thought about meaning and semantics. It is therefore of interest to envisage an argument to the effect that the early Russell’s use of the concept of truth cannot be substantial; or that he had deep theoretical reasons to be sceptical about substantial uses of a truth-predicate. An argument of this general shape can be found in Thomas Ricketts’ writings on Frege.170 Ricketts is concerned with Frege’s logic and an anti-semantical argument that he claims to find in Frege’s writings, but an analogous argument can be attributed to the early Russell as well, although I am not aware that anyone has done this attribution so far.171

4.5.7.2 An Analogy with Frege

In interpreting Frege about logic, Ricketts claims that a certain anti- semantical argument – more specifically, an argument about truth – holds the key to Frege’s thinking about logic. This argument is the most important reason why Ricketts resists what he sees as the standard interpretation of Frege as the founding-father of a modern, semantic conception of logic. The starting-point of the interpretive argument is an interpretation of Frege’s anti-psychologism which gives a central role to the notions of judgment and assertion. This in turn gives rise to an argument against the definability of truth; this is the well-

170 See Ricketts (1986, sec. II), (1996) and (1997). 171 Although the rest of the section is mainly about Russell, I am indebted to Shieh’s recent, very informative and enlightening discussion about Ricketts’ argument (Shieh 2001). Ch. 4 Logic as the Universal Science I 443 known, if not very well-understood, infinite regress or circularity argument whose fullest exposition is to be found in Frege’s late essay Der Gedanke. On Ricketts’ reading, Frege takes this argument to show that truth is not a property of thoughts. Given this result, Ricketts reaches the ultimate conclusion of the argument: since, for Frege, truth is not a property of thoughts, he could not have had a genuine semantic theory. On the face of it, the third part of the argument is the least problematic (cf. Shieh 2001, p. 100). If there is no such thing as truth as a property, there is no such thing as ascribing truth to anything, including Fregean thoughts. If there are assertions that seem to involve such ascriptions, these must be explained away; and Frege has at least the beginnings of a theory of how this is to be accomplished, namely a series of remarks on assertoric force Since semantics presupposes the legitimacy of truth-ascriptions, it follows that semantics is, strictly speaking, an impossible enterprise.172 The application of this interpretive argument to Russell is fairly straightforward; we need only slightly modify the first premise and conclusion. Instead of an anti-psychologistic argument based on a particular notion of judgment and assertion, Russell has his peculiar conception of true proposition as a metaphysically fundamental notion.173 Given this conception, one can – and Russell does – develop an argument to the effect that truth is not really a property of primary truth-bearers. This result yields, not an anti-semantical conclusion in the sense of strict semantics, but the conclusion that truth qua property and attributions of truth to propositions cannot play a central role in an account of logic. In other words, since truth and falsity are not properties of primary truth-bearers, such locutions as ‘p is true’

172 Given this conclusion, Ricketts must tell an alternative story about what Frege is doing in the early sections of Grundgesetze. Ricketts use the notion of “elucidation” to this effect; Ricketts (1997) is his fullest treatment of the subject. 173 Cf. section 4.4.6.4, where the notion of true proposition was discussed. 444 Ch. 4 Logic as the Universal Science I and ‘q is false’ as well as their generalisations cannot be accepted at face value. In sum, the analogy with Frege suggests that we apply the following interpretative argument to the early Russell:

(1) Russell’s conception of propositions as metaphysically fundamental entities; (2) an argument to the effect that, given (1), truth cannot be regarded as a property of propositions; (3) given (2), it follows that neither truth qua property nor truth- ascriptions can play a substantial, explanatory role in logic.

To repeat, the conclusion of this argument does not hinge on whether discourse featuring the truth-predicate should be classified as semantic or not. The present concern is simply what conclusions we should draw about the early Russell’s use and notion of truth. We have already met the fundamentals of the first premise of this argument, to wit, the special, metaphysically constitutive role that the notion of true proposition plays in the early Russell’s metaphysics. Recall the following Principle of Truth:

(PT) a is F if and only if the proposition /a is F/ is true.

(The slashes, it will be recalled, are used to denote a Russellian proposition.) That is to say, an object’s having a property or standing in a relation to other objects amounts to there being a suitable array of true propositions. (PT) is thus more than just a (necessary) equivalence. It does not state that, for every fact, there is a corresponding true proposition. What (PT) is meant to capture is the metaphysically constitutive role that Russell’s theory assigns to propositions: a is F because there is a certain proposition, describable in a specific way, that is true; and here ‘because’ is to be considered explanatory in the sense, whatever it is, that is appropriate in metaphysics. To the best of my knowledge, the early Russell nowhere explicitly formulates (PT) or any other equivalent principle. Its attribution to Ch. 4 Logic as the Universal Science I 445 him can nevertheless be justified on the grounds that it enables us to make sense of an argument against truth-definitions that is an essential ingredient in his thinking about truth. This argument brings us to the second step of the overall argument we wish to construct. An early version of Russell’s argument against truth-definitions can be found in the 1899-manuscript, The Fundamental Ideas and Axi- oms of Mathematics.174 The gist of the argument is repeated in a more tractable form in (Russell 1905c), and it is this formulation that I shall discuss here. Having first considered and rejected a possible interpretation of correspondence, Russell writes:

But even supposing some other definition of correspondence with reality could be found, a more general argument against definitions of truth would still hold good. An idea is to be true when it corresponds with reality, i.e. when it is true that it corresponds with reality, i.e. when the idea that it corresponds with reality corresponds with reality, and so on. This will never do. In short, if we don’t know the difference between a proposition’s being true and not being true, we don’t know the difference between a thing’s having a property and not having it, and therefore we can’t define a thing as true when it has a certain property such as corresponding with reality. (Russell 1905c, pp. 493-494)

Ricketts (2001) calls this passage “obscure”. Applying (PT) to it, however, makes Russell’s argument reasonably straightforward. In formulating the argument, Russell uses the correspondence theorist as his explicit target, apparently because the concept of correspondence continues to underlie idealists’ thinking about truth and ‘ap- peals most to the plain man’ (1905c, p. 492). Since nothing in Rus- sell’s argument depends upon the exact formulation of the idea of correspondence, the objection may be formulated in terms of the following undifferentiated notion of correspondence:

(1) p is true if, and only if, p corresponds with reality.

174 See Russell (1899c, p. 285). 446 Ch. 4 Logic as the Universal Science I

“p corresponds with reality” appears to be an ordinary predication; the right-hand side of the equivalence attributes a certain relational property to an entity. Thus, the theory purports to explain what truth is by capturing the property that truth consists in. The next step is the crux of Russell’s argument. I propose the following reconstruction, which helps to make sense of Russell’s claim that “if we don’t know the difference between a proposition’s being true and not being true, we don’t know the difference between a thing’s having a property and not having it”. Since “p corresponds with reality” is an instance of “entity e has property X”, (PT) is brought to bear on it; this yields:

(2) p corresponds with reality if, and only if, it is true that p corresponds with reality, or, following our stipulation,

(2´) p corresponds with reality if, and only if, the proposition /p corresponds with reality/ is true.

By the correspondence theorist’s lights, the right-hand side of (2), or (2´), should be analysed with the help of the notion of correspondence. This yields:

(3) it is true that p corresponds with reality if, and only if, the proposition that p corresponds with reality corresponds with reality, or (3´) /p corresponds with reality/ is true if, and only if, /p corresponds with reality/ corresponds with reality.

Again, understanding the right-hand sight of the purported equiva- Ch. 4 Logic as the Universal Science I 447 lence brings us back to the notion of truth, and if we continue to apply the correspondence theorist’s definition of truth, we embark on an infinite regress.175 Hence, according to Russell, the attempted definition of truth in terms of correspondence generates an infinite regress. Accordingly, the original definition, (1), must be written off as a failure. Although the argument is explicitly targeted on truth as correspondence, it is in fact perfectly general, as Russell himself points out in the above quotation. “Corresponds with reality” is a mere place- holder; since nothing in the argument depends upon its substantial content, it may be replaced by any other purported explication of truth. The only thing that matters for Russell’s argument is the general or schematic form of truth-definitions. Here the presupposition is that truth is a genuine property of truth-bearers. The real target of the argument is therefore the schematic truth definition, i.e., the idea that truth consists in some property possessed by propositions. Not only truth as correspondence but any definition of truth that represents truth as a property leads to a vicious infinite regress, according to this argument.176 177

175 The other possibility would be to press the charge of circularity. In this case the argument would simply be that since (PT) shows that the notion of true proposition is more fundamental than that of fact, it would be viciously circular to try to reduce the former to some fact about a proposition. 176 Since Russell’s argument is based on the identification of facts with true propositions, the above argument is in fact restricted to definitions of truth which identify the truth of propositions with “something else”; thus its real target is not truth simpliciter, but, rather, the notion of a true proposition. Since however, Russell also accepts that any other entity than proposition can be said to be true only because it bears a suitable relation to a fully determinate proposition, this restriction turns out to be vacuous. 177 Comments from Nicholas Griffin made me to see that the above interpretation of the early Russell’s conception of truth is not in fact the only possible. For one could read him arguing merely for the conclusion that truth is an indefinable, primitive property; on this view, since the notion of truth is metaphysically basic, all other kinds of predication reduce, metaphysically 448 Ch. 4 Logic as the Universal Science I

4.5.7.3 Frege’s Version of the Argument against Truth- definitions

I have already remarked that Russell’s reasoning bears close resemblance to a much better-known argument which Frege deployed to undermine definitions of truth.178 Consideration of the latter enables us to throw further light on the details of Russell’s argument. There is no consensus among Frege-scholars on how the argument should be reconstructed.179 Its gist, however, appears to be the claim that once truth is identified with some property like correspondence, the further question whether a particular content, , is true is to be resolved by the application of the proposed definition to . Frege holds, however, that there is an exceedingly close connection between truth, on the one hand, and judgment or assertion on the other, a connection that is captured by the following principle:

(*) To judge is to acknowledge the truth of a thought.180 speaking, to the one case where truth is predicated of propositions. The view that I have attributed to Russell is the more radical view which purports to dispense with primitive predication altogether; property possession reduces to the truth of a proposition, but the latter is not itself a species of property possession. This view is, admittedly, somewhat radical, and the early Russell often speaks of truth and falsity as if they are properties of propositions; see, for example, the end or Russell (1904a). The more radical view nevertheless receives support from Russell’s presentation of the regress-argument in his (1905c), and I shall assume it was his considered view, admitting that the issue may be less clear-cut than is indicated in text. I am grateful to Tuomo Aho for a discussion on this point. 178 In both cases the explicit target of the argument is the correspondence conception of truth, but both arguments are intended to apply generally. For Frege’s formulation, see Frege (1918-19a, pp. 352-3). 179 The most convincing reconstruction is given by Greimann (1994) and (2003, pp. 205-213). See also Kemp (1995) for an unorthodox discussion of Frege’s conception of truth. 180 See Frege (1918-19a, p. 355-356), where this principle is explicitly stated. Ricketts (1995, p. 131, fn. 25) is clearly correct, when he says that (*) Ch. 4 Logic as the Universal Science I 449

Given (*), it follows, according to Frege, that the determination of the truth-value of is an impossible task: in order to perform this task, one would have to put oneself in a position to judge, or assert, that the proposed definition of truth applies to , i.e., that the content has the property which the definition assigns to truth-bearers. But the judgment that has a property ƶ amounts to the acknowledgment that the thought that is ƶ is true. In order to answer the original question, then, one would first have to establish the truth of another thought. But this in turn presupposes exactly the same procedure; hence, an attempt to resolve the original problem leads to an infinite regress or else is rendered impossible on account of having been specified in a manner that is viciously circular. It is evident enough that Frege’s argument, insofar as is understood in the way just indicated, misrepresents the correspondence theorist’s commitments.181 On any reasonable conception of truth as correspondence, the determination whether a given content is true does not take the form of a straightforward application of the proposed definition of truth to a particular case: to determine whether the content is true, for example, we do not compare the content with reality to see whether the required correspondence is there or not; rather, we answer the question simply by conducting appropriate investigations, which put us in a position to assert that snow is or is not white. Frege’s argument, by contrast, is based on the presupposition that the act of asserting, or judging, and the act of recognizing that a thought (content) is true are in fact one and the same act; this is precisely what is stated in (*) – it is from this assumption that Ricketts starts to develop his interpretative argument. To this somewhat esoteric-sounding claim the reply may be made on behalf of the correspondence theorist that the two acts are, indeed, two and hence separate from one another. Ontologically speaking, correspondence theory is grounded in the assumption that the truth of a content is one expresses a “stable element in Frege’s conception of judgment and truth”. 181 The following two paragraphs draw heavily on Kemp (1995). 450 Ch. 4 Logic as the Universal Science I thing (entity, fact), whereas the condition which determines the content as true or false is another thing (entity, fact). Frege’s argument, by contrast, is grounded, precisely, in the denial of this assumption. On a plausible reading, the core of Frege’s theory of judgment is the identification of content with the truth-value conferring condition. Hence it is natural to think of his argument against truth- definitions as spelling out a particular consequence of that theory; since facts are (that is, are identical with) true thoughts, truth itself cannot be explained by referring to some further fact which could be cited as constituting the truth of a thought.182

4.5.7.4 The Metaphysics of Truth

Be these details as they may, Russell’s version of the no truth definition-argument is best construed in this way, as a metaphysical regress argument which is grounded in (PT). This principle ensures that the fact-truth connection, being one of identity, is so tight that the truth of a proposition cannot be understood as a kind of fact, or as a property possessed by a proposition. Thus, the charge that the regress in

182 For an interpretation of Frege’s argument along these lines, see Kemp (1995, pp. 39-42). According to Kemp, the correspondence theorist’s reply to Frege assumes a conception of thought that is entirely foreign to him (id., p. 39). For Frege, facts or states of affairs are not something that thoughts represent, or fail to represent; rather, facts are true thoughts (Frege says so explicitly at (1918-19a, p. 368). Kemp’s line of thought is a plausible one, but there is nevertheless some difficulties standing in its way. Above all, it is not easily reconciled with Frege’s semantics – I ignore here the complication that the above line of thought could be used as one reason to resist the attribution of genuine semantics to Frege. When he does semantics, he explains that thoughts are senses of sentences, and senses in turn are ways in which referents are given. This notion of sense, it seems to me, is more naturally associated with the view that associates propositions with representations, rather than with the conception that underlies the identity theory. I do not know how these two standpoints should be reconciled. Ch. 4 Logic as the Universal Science I 451 which purported definitions of truth are caught is vicious rests on the specific sense which Russell attaches to (PT), the principle generating the regress. If the regress were just a consequence of a recognition that, correlated with each predication, there is a predication of truth, it would be entirely harmless. (PT), however, must be taken as an explanatory principle; it is supposed to deliver an analysis of the notion of fact, or of what it is for something to be the case. This argument can be further clarified by comparing it with the reasoning that Russell applies to Bradley’s regress in the Principles.183 In discussing Bradley’s regress, he draws a distinction between two kinds of regresses, and claims that only one of them is vicious:

(A) regress of meaning; (B) regress of implication.

The first regress is of the sort that may arise in the analysis of propositions. That Russell calls it a regress of meaning may be mildly surprising; after all, questions of meaning in any ordinary sense are not on Russell’s agenda at this time, and what he means by “meaning” when applied to propositions he nowhere really explains. It is nevertheless quite evident that by the “meaning” of a proposition he refers to its constituents; hence, one is concerned with the meaning of a proposition when one tries to identify its constituents, i.e., analyse the proposition. Recall Russell’s claim that Bradley’s regress is blocked by the observation that it is merely one of implication and not of meaning. Given a proposition like /aRb/, its unity – that which distinguishes it from a mere list of terms – is guaranteed by the relation R, which occurs as a relating-relation in the proposition (as we saw in discussing the problem of unity, this notion of relating-relation is an obscure one; since it constitutes Russell’s real response to Bradley; it is only if the notion is a viable one that Russell has a reply to Bradley; this,

183 See above, section 4.4.6.2. Russell discusses Bradley’s regress in the Principles, §55 and §99. 452 Ch. 4 Logic as the Universal Science I however, is not to the present point). He admits that further relations between the original proposition’s constituents are implied; for instance /aRb/ implies propositions like /aR´R/, /bR´R/, and so on. He holds, however, that these further relations are not constituents of the original proposition, and do not show up in the analysis of /aRb/. They are therefore not part of the meaning of the original proposition, and the ensuing regress – for instance, that /aRb/ implies /aR´R/, which in turn implies /aR´´R´/, and so on – is entirely harmless. Russell’s reasoning is readily applied to the present regress argument. Here the claim is that if the correlation between predication and truth were merely one of implication – /a is F/ implies /it is true that a is F/, which implies /it is true that it is true that a is F/, and so on – it would be inconsequential. This kind of regress is a natural ally of the conception of content and fact which goes together with the correspondence theory of truth. Associated with the fact that snow is white, there is an infinity of ever more complex contents (, , , and so on), but each fact to the effect that a content is true remains distinct from the truth-determining fact. As I read him, however, Russell holds that a truth is not merely implied by a thing’s having a property or standing in a relation. Truth may not – in fact does not – occur among the constituents of a proposition, once the proposition has been analysed, and to that extent the analogy with Bradley’s regress is incomplete. Nonetheless, a thing’s having a property and standing in a relation is metaphysically dependent on, or determined by, there being suitable true propositions; this is the import of (PT). Hence any attempt to analyse truth in terms of some property or relation generates an explanatory circle, or infinite regress. This comes close to the conclusion that Russell himself draws from the regress-argument: “if we don’t know the difference between a proposition’s being true and not being true, we don’t know the difference between a thing’s having a property and not having it”. That is, the former distinction is more fundamental than the latter; and Ch. 4 Logic as the Universal Science I 453 here, I take it, “more fundamental” is best construed as “metaphysically more fundamental”. It follows that “we can’t define a thing as true when it has a certain property such as corresponding with reality.” We can’t, because understanding the definiens would assume a prior grasp of the definiendum.184

4.5.7.5 Truth-Primitivism and Truth-Attributions

What was said above means, also, that the Principle of Truth is a potentially misleading formulation of the metaphysical dependence it is supposed to capture; the surface-form of the right-hand side of (PT) suggests that the truth of a proposition is itself a case of property- possession. But this cannot be, on pain of rendering the Principle viciously circular. The early Russell’s conception of truth is best characterised as a combination of two ideas. Firstly, there is what may be called truth- primitivism. Since there cannot be a definition of truth in the fundamental sense of “true proposition”, one can at best enumerate the salient features of this concept. These would include at least the following: (1) truth and falsity belong to propositions, although they should not be considered their properties; (2) every proposition is true or false; (3) no proposition is both true and false; (4) truth is stable, i.e., a proposition that is true is always true (or eternally true), and to think otherwise is to fail to distinguish between propositions and sentences.185

184 In spite of Russell’s formulation, I do not believe that his point is primarily about understanding or explanation. As above, we might rephrase the point of the argument saying that an attempt to analyse propositional truth in terms of some property generates a metaphysical regress. There is, though, nothing specifically wrong with Russell’s own wording; after all, some things that call for an explanation are metaphysical, and hence there is room for metaphysical explanations and explanations in metaphysics. 185 Truth-primitivism is, in the form Russell gives to it, closely related to the so-called identity theory of truth that has attracted some attention lately: 454 Ch. 4 Logic as the Universal Science I

Secondly, to truth-primitivism must be added the sui generis nature of truth-attributions. In the Principles Russell makes an attempt to explain what truth is by forging a connection between it and the notion of assertion; this move, I take it, is as close as he comes in the Principles to recognizing that propositional truth is not an instance of predication, the conclusion which we reached by considering the Principle of Truth. The conclusion may sound familiar enough; a similar-sounding tactics is found in Frege, for instance.186 The feeling of familiarity begins to disappear, however, when one finds Russell arguing, in §52 of the Principles, that the notion he has in mind is assertion in logical rather than psychological sense, where the latter would be, roughly, the down-to-earth notion that people have in mind when they speak about “assertion”. What the logical sense is, Russell never succeeds in explaining. It is the notion that is supposed to distinguish truth from falsity (ibid.), and these two are further connected with the difficult concepts of logical subject and propositional concept as well as with the equally problematic distinction between internal and external relations. No coherent conception of the connection between truth and assertion emerges from these remarks, and Russell himself concludes, clearly dissatisfied with his own discussion that since “[t]he nature of truth [...] belongs no more to the principles of mathematics than to the principles of everything else”, no detailed discussion of the concept of truth is called forth (ibid.) The notion of assertion, then, does not really help to illuminate the concept of truth.187

see Candlish (1999), (2003), (2007, Ch. 4); Dodd (2000). 186 For an exposition of Frege’s theory of assertion, see Greimann (2003, Ch. 4). 187 As we have seen (see section 4.4.7), explaining truth is not the only reason for Russell to focus on the notion of assertion. There are at least two other such reasons: the problem of unity and the theory of logical inference. Of these three reasons, the last seems like a cogent one, whereas the first two are patently less so. Ch. 4 Logic as the Universal Science I 455

4.5.7.6 Use of the Truth-Predicate

Bracketing questions about the difficult notion of assertion in logical sense, we have the following conclusions about the early Russell’s conception of truth:

(1) truth in the fundamental sense (propositional truth) cannot be defined; (2) truth in the fundamental sense is not to be conceived of as a property of propositions; the second conclusion, in turn, gives rise to a third:

(3) there are no ineliminable uses of a truth-predicate, when truth in the fundamental sense is concerned.

It is the first and third conclusion that could be used to argue that the early Russell was in principle debarred from a “substantial use” of the concept of truth in logical theory. This is precisely the sort of conclusion that Ricketts draws in Frege’s case. According to Ricketts, Frege’s conception of judgment – essentially, the principle (*) discussed in section 4.5.7.3 – does not permit any real metaperspective (1986, p. 79). Hence, also, Frege cannot have a definition of truth (cf. (1) above) or ineliminable use of a truth-predicate (cf. (2) and (3) above). Ricketts, though, does not really discuss truth-definitions at all but concentrates on the second conclusion. Consider truth-definitions first. In the Principles there is no discussion of whether truth can be defined. However, Russell does defend an indefinability claim for implication, based on the view that since definitions state mutual implications, implication itself cannot be defined. A proposed definition introducing the notion would itself contain implications, and would therefore be circular somewhat in the manner that a definition of truth would be, according to the regress 456 Ch. 4 Logic as the Universal Science I argument. Russell’s view on implication is in a marked contrast to a more modern view. Equipped with a distinction between object- and meta-languages, we would simply say that a definition of implication is a definition of an object-language expression, the definition itself being given in a metalanguage; and if the definition uses implications, that does not make it circular, for implication in the meta-language is not what is being defined. This suggests that Russell’s thinking about the fundamental concepts of logic – including implication and truth – is set up against a background of an absolutist conception of propositions, rather than within a framework of a multiplicity of languages. It does not follow from this, however, that his conception would be somehow intrinsically inimical to a definition of truth. For a universalist could well argue that a definition of truth is applicable to a calculus of logic, rather than to logic in the fundamental sense in which its notions (implication, truth, and so on) are not tied up to this or that language or calculus. From Russell’s perspective, such a definition would presumably presuppose the more fundamental notion of propositional truth, but that would be no objection, once it is made clear what the definition and its subsequent uses are supposed to accomplish. None of this is meant to suggest that the early Russell ever thought that a formal definition of truth was an important logical technique. Such finesse were unquestionably beyond his logical horizon; the point is merely that his understanding of truth as an indefinable, metaphysical notion is not enough to rule out the possibility of a formal truth-definition. Consider, next, the third conclusion, to wit, that a universalist attracted by the regress argument cannot have ineliminable uses of a truth-predicate, because such uses flatly presuppose that truth in the fundamental sense is a property. On the face of it, this conclusion is contradicted by Russell’s practice, for it seems that he does use a truth- predicate in the Principles. If this is correct, anyone who wants to defend the conclusion should be able to explain away these cases. The simplest and most prosaic way out would be to argue that in Ch. 4 Logic as the Universal Science I 457 the Principles Russell had not yet seen the implications that “propositional truth” has for truth-attributions. This explanation is not without plausibility; particularly so, when one takes into account the modest nature of these attributions. In §1, after giving a list of “logical constants”, Russell has the following to say about truth: “[i]n addition to these [sc. the logical constants], mathematics uses a notion which is not a constituent of the propositions which it considers, namely the notion of truth”. It appears, then, that mathematics and logic do have a use for the notion of truth. But perhaps the cases where a truth- predicate is being used can be ignored on the grounds that when Rus- sell uses “truth” and “falsity” as predicates on propositions, this is no more than informal, or perhaps propaedeutic talk that should not be construed sensu stricto. This suggestion, it must be admitted, is not without plausibility; looking for uses of a truth-predicate in the Principles, they are mostly not there, or when they are, they appear to be very modest indeed: see, for example, §16, a part of which has already been quoted. Another, more intriguing possibility would be to argue that there are, in fact, two notions of truth at work in the Principles: firstly, there is the fundamental notion, which is that of propositional truth (and which is connected in some way with assertion); secondly, there is another notion, one which is expressed by a truth-predicate and is construed as a predicate on propositions (or, possibly, on sentences expressing propositions). This second notion would be parasitic upon the first, since every predication is metaphysically dependent upon there being a suitable proposition that is, so to speak, primitively true; but, one could argue, it is precisely this second notion that Russell can use for the purposes of modest truth-attributions. Be that as it may, there appears to be at least one instance where Russell uses a truth-predicate in a manner that is genuinely ineliminable and cannot be explained away as merely propaedeutic or elucidatory. This is Axiom (4) of the Propositional Calculus (see §18 of the Principles), which reads as follows: 458 Ch. 4 Logic as the Universal Science I

(4) A true hypothesis in an implication may be dropped, and the consequent asserted

Axiom (4) is not an auxiliary statement, explaining this or that feature of the calculus, thus helping a potential reader to understand what the calculus is all about. Its role is to vouchsafe the correctness of the inferences that can be performed in the calculus, this being part of the reason why the calculus deserves to be called a calculus for logic. If “is true” was really excluded as a predicates on propositions, (4) would have to be written off as illegitimate. But surely Russell cannot think that the axioms of his calculus include one that should not really be there. What he does say is that this principle “is incapable of formal symbolic statement, and illustrating the essential limitations of formalism” (ibid.) The point of this remark, I take it, is that the “axiom” or “principle” that legitimates transitions from propositions to propositions – a rule of inference, that is – is not an ordinary proposition on a par with the rest of the axioms; if it were, a regress of the kind that Lewis Carroll described would destroy the very possibility of drawing conclusion from given propositions.188 Russell’s point is therefore simply that there is a difference between axioms and rules of inference (these are not his terms, but the point is the same). It is tempting to reformulate this and say that Rus- sell is here gesturing towards an explicit recognition of a metaperspective.189 And it is significant that in explaining the point Russell uses expressions like “symbolic statement” and “formalism”, indicating that, even for a universalist, a “meta-perspective” is a matter that has to do with a calculus for logic, point that we have already made on more than one occasion. Russell’s point is thus not about what is and what is not legitimate per se, but what can and cannot be expressed

188 See Carroll (1895). Cf. Russell’s discussion of this point in §38 of the Principles; for further discussion, see section 5.9.3 below. 189 This is how Landini (1998, p. 45) puts it. Ch. 4 Logic as the Universal Science I 459 within a calculus.190 Summing up the discussion, Russell holds that truth in the fundamental sense of propositional truth is a non-semantic, metaphysical notion that cannot be defined and cannot be regarded as a property of propositions. It does not follow from this, however, that the concept could not in principle play a significant, even explanatory role in logic. This role, however, concerns calculi for logic, rather than logic in the fundamental sense which is not captured by this or that calculus.

4.6. Russell’s Conception of Mathematical Theories

4.6.1 General Remarks

Russell’s conception of logic should also be considered in the context of its use, i.e., by considering his understanding of the proper content and formulation of mathematical theories. Returning to the notion of interpretation, it would be most surprising, from this point of view, had he simply written it off as illegitimate.191 The idea that mathematical theories are capable of a multiplicity of interpretations (models, applications, etc.) was becoming commonplace among mathematicians of the late 19th century. Russell, under the influence of Peano, became a vigorous advocate of “modern mathematics”, one of whose defining characteristics was precisely this abstract perspective on mathematical theories. This fact alone should make us extremely

190 In considering the case of truth-definitions, we seemed to arrive at the conclusion that Russell tends to think about logic in absolutist terms. Here, by contrast, it appears that the notion of a calculus for logic emerges as the real point of interest. This is perhaps a natural outcome of the universalist logician’s conceptual situation, as he has room for both conceptions; a calculus (or calculi) for logic that presupposes logic in the absolute sense, or logic that is not relativized to this or that calculus, or language. 191 For the generic notion of interpretation, see above, section 4.5.5. 460 Ch. 4 Logic as the Universal Science I sceptical of any reading of Russell’s conception of logic which implies that he simply dismissed ‘alternative interpretations’ and similar ideas. It was by no means easy for mathematicians and philosophers to draw the appropriate methodological and philosophical lessons from the development of the abstract conception of mathematical theories, and several, often conflicting reactions were eventually suggested: conventionalism, formalism, deductivism, some varieties of logicism, structuralism and model-theoretic thinking are all direct descendants of the abstract viewpoint.

4.6.2 The Frege-Hilbert Controversy

One episode relating to the philosophical interpretation of the abstract viewpoint is the Frege-Hilbert controversy concerning the foundations of geometry, i.e., concerning how best to understand the different ingredients that together go on to make up an axiomatized geometry. This exchange is well-known and has been analyzed in considerable detail in the secondary literature.192 What is less well- known is that at about the same time a similar debate, albeit on a smaller scale, took place between Poincaré and Russell. We gain a useful insight into the logicist Russell’s views on mathematical theories if we (i) review the essential content of the Frege-Hilbert debate, (ii) relate this to the Russell-Poincaré debate, and (iii) consider against this background the logicist conception of mathematics which is found in the pages of the Principles. According to Frege, there is an absolutely sharp distinction between axioms and definitions. An axiom is a truth that is fundamental to a discipline (like Euclidean geometry). This foundational character of an axiom shows itself in the facts that an axiom must be self-evident

192 See, for example, Resnik (1980, pp. 106-119), Hintikka (1988), Coffa (1991, pp.135-7), Demopoulos (1994), Hallett (1994), Shapiro (1997, pp. 161-70). Ch. 4 Logic as the Universal Science I 461 and such that it neither needs nor admits a proof.193 A definition is a stipulation to the effect that an expression that previously had no meaning means and therefore designates the same entity as the definiens. A definition thus contains a sign which lacks antecedent meaning and denotation. By contrast, sentences expressing axioms must contain no expression whose meaning is not fully determinate. If an axiom-sentence did contain such an expression, it would not express a thought and could not be true. Frege thus assumes that an axiomatized theory deals with or is about some specific subject- matter, like space or the natural numbers, subject-matter which is given independently of the axioms of the theory. Semantically speaking this givenness is shown by the fact that the meanings, and hence denotations, of the primitive expressions of the theory – expressions which can be used in definitions but are not themselves defined – must be graspable independently of grasping the content of the relevant axioms. Frege’s conception of axiomatics was traditional and conservative; some would even call it reactionary, when it is considered against the background of the relevant historical facts.194 It is a conception that is radically different from the views found in Hilbert’s new geometry. Hilbert does claim, in the short Preface to his Grundlagen der Geometrie, that the geometer’s task amounts to a logical analysis of our spatial intuition, a description that would have fitted quite well what Frege thought about the matter. As commentators have pointed out, however, intuition in fact plays no role in Hilbert’s actual setting up of Euclidean geometry, or at any rate, that its role is limited to “motivation and heuristic”, as Shapiro (1997, p. 157) puts it. Paul Bernays has given the following summary of Hilbert’s approach under the title “abstract axiomatics”:

193 See Burge (1998b) for a detailed discussion of Frege’s conception of axioms. 194 See, for instance, Freudenthal’s condescending remarks on Frege at the beginning of his (1962). 462 Ch. 4 Logic as the Universal Science I

A main feature of Hilbert’s axiomatization of geometry is that the axiomatic method is presented and practiced in the spirit of the abstract conception of axiomatics that arose at the end of the nineteenth century and which has been generally adopted in modern mathematics. It consists in abstracting from the intuitive meaning of the terms for the kinds of primitive objects (individuals) and for the fundamental relations and in understanding the assertions (theorems) of the axiomatized theory in a hypothetical sense, that is, as holding true for any interpretation or determination of the kinds of individuals and of the fundamental relations for which the axioms are satisfied. Thus, the axiom system is regarded not as a system of statement about a subject-matter but as a system of conditions for what might be called a relational structure. Such a relational structure is taken as the immediate object of the axiomatic theory, its application to a kind of intuitive object or to a domain of natural science is to be made by means of an interpretation of the individuals and relations in accordance with which the axioms are found to be satisfied. (1967, p. 497)

Given Frege’s tradition-bound views on axiomatic theories, it is not hard to understand why he failed (initially) to grasp the point behind Hilbert’s Grundlagen. For Frege, abstract axiomatics involves a misunderstanding of both the nature of axioms and the nature of definitions. The essence of Frege’s own views consists in the following two theses:

Ontological thesis: axioms are (self-evidently) true, and they are true in virtue of a subject-matter.

Semantic thesis: definitions give meanings and fix denotations; ultimately the fixity of meaning is dependent upon primitive terms, whose meanings and denotations are simply given (and must therefore be grasped if the content of axioms is to be grasped).

Frege was misled in these respects by Hilbert’s manner of presentation. At the beginning of §1 of Grundlagen, its author enjoins the reader to consider three distinct system of things, composed of entities that will be called “points”, “straight lines” and “planes”, respec- Ch. 4 Logic as the Universal Science I 463 tively; the reader is told furthermore that together these entities are called “elements of the geometry of space”, or more simply, “elements of space”. To an unguarded reader like Frege such a formulation may well suggest a fairly traditional approach, with the difference that Hilbert’s actual presentation raised logical rigour to a level previously almost unheard of among working mathematicians – but on this point Frege was in complete agreement with Hilbert. However, upon reading Hilbert’s explanation that, for example, his axioms of order “define the idea expressed by ‘between’”, an advocate of the traditional conception is likely to protest. Frege did just this, writing to his colleague that if he were to set up an axiom like “If A, B, C are points of a straight line and B lies between A and C, then B lies also between C and A”, he “would presuppose a complete and unequivocal knowledge of the meanings of the expressions “something is a point on a line” and “B lies between A and C”, etc. (Frege 1980, p. 37). That would also mean that this axiom cannot be regarded as a definition in Frege’s sense of the word. Frege suspects, furthermore, that Hilbert may be using “between” ambiguously, since elsewhere in the book a different meaning is given to the word (Frege is here referring to Hil- bert’s independence proofs). And this in turn indicates that the word does not have a meaning at all at its first occurrence in Hilbert’s axiom II.1. But if this is correct, Frege continues his protest, the axiom cannot express a fundamental fact about our spatial intuition. (ibid.) Hilbert responded to Frege’s queries, carefully explaining that his axioms constitute as full explanations or definitions of the relevant geometrical concepts as is possible:

I do not want to assume anything as known in advance; I regard my explanation in sect. 1 as the definition of the concepts point, line, plane – if one adds again all the axioms of groups I to V as characteristic marks. If one is looking for other definitions of a ‘point’, e.g., through paraphrase in terms of extensionless, etc. then I must indeed oppose such attempts in the most decisive way; one is looking for something one can never 464 Ch. 4 Logic as the Universal Science I

find because there is nothing there; and everything gets lost and becomes vague and tangled and degenerates into a game of hide and seek. (Frege 1980, p. 39)

In other words, even if Hilbert uses familiar words (“point”, “straight line”, “between”, etc.), these are, as we might say, freely reinter- pretable, and do not really possess, in that sense, a fixed meaning and denotation; as Hilbert puts it, knowledge of these is not assumed “in advance”. Hilbert’s axioms therefore apply to any “system of things” that satisfies the relevant axioms. This is also something that Hilbert tried to convey to Frege:

You say that my concepts, e.g. ‘point’, ‘between’, are not unequivocally fixed; e.g. ‘between’ is understood differently on p. 20, and a point there is a pair of numbers. But it is surely obvious that every theory is only a scaffolding or schema of concepts together with their necessary relations to one another, and that the basic elements can be thought of in any way one likes. If in speaking of my points I think of some system of things, e.g. the system: love, law, chimney-sweep... and then assume all my axioms as relations between these things, then my propositions, e.g. Py- thagoras’ theorem, are also valid for these things. In other words: any theory can always be applied to infinitely many systems of basic elements. One only needs to apply a reversible one-one transformation and lay it down that the axioms shall be correspondingly the same for the transformed things. This circumstance is in fact frequently made use of, e.g. in the principle of duality, etc. and I have made use of it in my independence proofs. (Frege 1980, pp. 40-41)

This explanation by Hilbert was apparently helpful, for after an initial period of bewilderment, Frege began to appreciate (at least up to a point) the guiding idea behind Hilbert’s new geometry. This is shown by the very next letter he penned to his colleague:

I believe [...] I can see a little more clearly what your plan is. It seems to me that you want to detach geometry entirely from spatial intuition and to turn it into a purely logical science like arithmetic. The axioms which are usually taken to be guaranteed by spatial intuition and place at the base of the whole structure are, if I understand you correctly, to be car- Ch. 4 Logic as the Universal Science I 465

ried along in every theorem as its conditions – not of course in the fully articulated form, but included in the words ‘point’, ‘line’, etc.” [...] By placing yourself in a higher position from which Euclidean geometry appears as a special case of a more comprehensive theoretical structure, you widen your view so as to include examples which make the mutual independence of those axioms evident. Although I am struck here by a doubt, I will not pursue it further here. The main point seems to me to be that you want to place Euclidean geometry under a higher point of view. (Frege 1980, pp. 43-44)

This insight – “you want to detach geometry entirely from spatial intuition and to turn it into a purely logical science like arithmetic” – is developed further in a series of papers on the foundations of geometry that Frege published in 1903 and 1906.195 There Frege shows in some detail how Hilbert’s innovations could be given satisfactory formulations within a logical framework that Frege himself found acceptable. Frege’s results need not be recapitulated here, since many of his points are also present in Russell’s works, to which we shall turn next.

4.6.3 The Russell-Poincaré Debate

We have already mentioned that Poincaré and Russell entered a brief exchange of mutual criticisms on the nature of geometric knowledge that is broadly analogous to the much better-known controversy between Frege and Hilbert. Russell had published his Essay on the Foun- dations of Geometry in 1897. At that time he did not have available to himself a logical framework comparable to Frege’s concept-script. There are nevertheless important similarities between their respective views on axiomatization, similarities that are due to the fact that both accepted the traditional, Euclidean conception of axiomatic theories. Poincaré published a review of Russell’s book in 1899, focussing, among other things, on Russell’s views about geometrical primi-

195 See Frege (1903a), (1903b) and (1906). 466 Ch. 4 Logic as the Universal Science I tives.196 Like Frege, Russell had expressed the view that geometric primitives are meaningful expressions, and their meanings and denotations are grounded in a domain of entities that is extraneous to the axiomatisation. Like Hilbert, Poincaré was thoroughly opposed to this piece of tradition, insisting that geometric primitives become devoid of meaning if we try to consider them in any way independently of the relevant geometric axioms. In his reply to Russell’s reply, Poincaré put the point as follows: “If one wants to isolate a term and exclude its relations with other terms, nothing will remain. This term will not only become indefinable, it will become devoid of meaning.”197 Thus, he presented the following challenge to Russell: “explain the primitives featuring in the axioms for projective geometry that you have given in your Essay!” Russell refused the challenge, arguing it was confused, thus making a point about definitions that is essentially identical with what Frege had initially tried to convey to Hilbert:

M. Poincaré demands ‘a definition of distance and the straight line, independent of (Euclid’s) postulate, and exempt from ambiguity and vicious circle’ [...]. He will perhaps be surprised at my replying that such a re- quest ought not to be made, since whatever is fundamental must be indefinable. I am convinced, however, that this is the only philosophically correct reply. As mathematicians almost invariably misunderstand the function of definitions, and as M. Poincaré appears to share this misunderstanding, I shall permit myself a few general remarks on the subject. A mathematical definition consists of any relation to some specified concept which is possessed only by the object or objects defined. In this sense, the projective straight line was defined above by its relations to points and planes. In this sense, also, the metrical straight line may be defined as a series of points which is wholly specified when two points belonging to the series are specified. [...] But such definitions are not definitions in the proper philosophical sense of the term. Philosophically, a term is defined when we are told its meaning, and its meaning cannot consist of relations to other terms. It will be admitted that a term cannot

196 See Poincaré (1899). 197 Poincaré (1900, p. 78); as translated in Stump (1994, p. 483). Ch. 4 Logic as the Universal Science I 467

be usefully employed unless it means something. What it means is either complex or simple. That is to say, the meaning is either a compound of other meanings, or it is itself one of those ultimate constituents out of which other meanings are built up. In the former case, the term is philosophically defined by enumerating its simple constituents. But when it is itself simple, no philosophical definition is possible. The term may still have a peculiar relation to some other term, and may thus have a mathematical definition. But it cannot mean this relation, and thus the mathematical definition becomes a theorem, which is true or false, and by no means arbitrary. [...] That some terms are indefinable must be evident after a moment’s reflection. For terms are defined by means of other terms, and to attempt a definition of all terms must therefore involve a vicious circle. Definition is exactly analogous to spelling. Words can be spelled, letters cannot. M. Poincaré’s question puts me in the unfortunate position of a schoolboy who is asked to spell the letter A, and forbidden to employ that letter in the process. If he be a mathematician, he will answer glibly: A is the letter before B; and if he is asked to spell B, he will say it is the letter after A. But if he knows what is meant by spelling, he will give up the task in despair. (Russell 1899a, pp. 409-11)

Russell uses rather strong words in defending the tradition against mathematicians. He nevertheless acknowledges that mathematicians are not simply confused but that their usage does not comply with the old use of “definition”, which is how definitions are used in philosophy; in addition to philosophical definitions, which purport to identify meanings, there are mathematical definitions, and these consist in the identification of an entity with the help of a relation the entity bears to a unique other entity or entities; in this sense we might, perhaps, define number 1 as the number that immediately pre- cedes 2, and 2 as the immediate successor of 1. Such a definition is at any rate suggested by the analogy Russell draws between defining and spelling.198

198 Although indicative of the basic idea, this example does injustice to the mathematician in an important respect. It would be easy to criticize these “definitions” from the standpoint of philosophical definitions on the grounds that, unless we know independently of their mutual relations what entities ‘1’ and ‘2’ stand for, such identifications-qua-definitions are really 468 Ch. 4 Logic as the Universal Science I

According to Russell, however, a mathematical definition or identification is not a proper, philosophical definition. It does not indicate the true meaning of the term in question and hence does not tell what the entity that the term stands for is intrinsically. And, evidently for him, the primitive terms of a theory must mean something in this further philosophical sense; otherwise they could not be “usefully employed”, as he puts it in the above quotation. Unfortunately, he does not explain how “can be usefully employed” should be understood, and he fails at this point to appreciate the point that, mathematically speaking, useful employment is a matter of axioms, rather than single expressions.199 However, when Russell further explains how mathematics and philosophy differ, he is not so much concerned with criticising the mathematical approach as emphasising the differences between the goals and methods of the two disciplines. As the following quotation shows, Russell is merely pointing out these differences, and he does not really explain what definition or analysis amounts to in the philosophical sense:

This difference [between mathematical and philosophical precision] may, I think, be described roughly as follows. In mathematics, what is important is the relations of terms. When two sets of terms have the same mutual relations, they are equivalent for mathematical purposes. What the terms are in themselves, is irrelevant; only their relations are important. But in philosophy, it is essentially the terms themselves that are important. We must ask, what do our terms mean? not, how are they related to other terms? Whenever a term is analyzable, philosophy should undertake the analysis. This is a task for which mathematical language is in general very ill-suited, and in which precision can only be attained with great difficulty. And even when precision has been attained, the meaning of fundamental terms cannot be given, but can only be suggested. If the useless. But to this the mathematician has an instant reply: what is being defined or identified are not really single entities at all (number 1, number 2, etc.) but what Hilbert referred to as “systems of things”, and what does the definition is the totality of axioms. 199 This was of course one of Hilbert’s main points against Frege. Ch. 4. Logic as the Universal Science I 469

suggestion does not call up the right idea in the reader, there is nothing to be done. Thus philosophical precision is a very different matter from mathematical precision. It is more difficult to attain, and far more difficult to communicate. And this is why the language of the theory of groups [which Poincaré had used] cannot help us to a philosophical account of the foundations of geometry. (Russell 1899a, pp. 411-2)

This passage is remarkable in that it shows that Russell was, despite his opposition to Poincaré, quite capable of recognising how far behind modern mathematics had left tradition. In this respect his attitude was more flexible – and more reasonable – than Frege’s initial reaction; Frege, after all, had initially discarded Hilbert’s work as failure, and did this in a manner that indicates that he had failed to grasp the gist of new geometry. J. Alberto Coffa and Stewart Shapiro have analysed the Russell- Poincaré exchange in some detail.200 Both have emphasised the resemblance that it bears to the much better-known Frege-Hilbert controversy. What both Coffa and Shapiro ignore, for the most part, however, is the fact that in Russell’s case commitment to the Euclid- ean tradition did not really last for long. Once we turn to the Principles to find out about its conception of mathematics, we can see at once that Russell has radically changed his mind, or at any rate that he is radically changing his mind; one remarkable thing about the philosophy underlying that book is that the Euclidean conception is, for the most part, simply missing. I say “for the most part”, because the work is not entirely consistent on this point. The presence of two such mutually incompatible standpoints – the traditional Euclidean view and the abstract conception of theories – is, again, best explained by referring to the contrast between Moorean and Peanist elements that runs through much that is philosophical in the Principles, as opposed to the technical development of logicism. The Moorean conception, with its emphasis on ‘intuition’ as the ultimate epistemological and semantic link between theory and reality, is clearly committed to a conception of axiomatic theories that is broadly similar to

200 Coffa (1992, pp. 120-22) and Shapiro (1997, pp. 161-170). 470 Ch. 4 Logic as the Universal Science I

Frege’s; on the other hand, most of the discussion of mathematics in the Principles accords with a conception of axiomatic theories that is calculated to fit “modern mathematics”, and here Moorean ideas are at best irrelevant and at worst positively misleading.

4.6.4 Philosophical and Mathematical Definitions

The distinction between philosophical and mathematical definitions which Russell had drawn in his reply to Poincaré thus recurs in the Principles, where it provides a particularly clear instance of the tension between the Euclidean and abstract standpoints. It is discussed briefly in §§16, 31 and 108. In the early sections Russell seems to construe the distinction in much the same terms as he had done in his exchange with Poincaré, and there is no indication that either side of the distinction would be intrinsically less valuable than the other. In §31 the point emerges in particular that entities which we have reason to accept as indefinables in the philosophical sense can nevertheless be given mathematical definitions, since definition in mathematics does not point out the entity in question but merely identifies it with the help of another entity, to wit, a propositional function, or according to Russell’s official theory, denoting concept.201 The “pointing out”, which Russell says is not effected by mathematical definitions, can only be accomplished by the philosophical method, which in the last analysis can only rely on “intuition”. §108, however, involves what appears to be radically different view of the relative merits of the two kinds of definitions:

Now definability is a word which, in Mathematics, has precise sense, though one which is relative to some given set of notions. Given any set of notions, a term is definable by means of these notions when, and only when, it is the only term having to certain of these notions a certain relation which itself is one of the said notions. But philosophically, the word

201 This is part of the reason why the notion of denoting was so important to Russell; see above section 4.4.9.2. Ch. 4. Logic as the Universal Science I 471

definition has not, as a rule, been employed in this sense; it has, in fact, been restricted to the analysis of an idea into its constituents. This usage is inconvenient and, I think, useless [...].

The passage does not allow us to decide whether Russell’s point about philosophical definitions is that they are, as such, useless, or whether that concerns their application to mathematics, only. For our present purposes, however, only the latter option is relevant: whatever else he may be doing in this passage, he is at least arguing that philosophical definition in the sense of analysing a complex into its constituents has no positive role to play in mathematics. Of course, as we saw in discussing Russell’s notion of proposition, “analysis” and “resolution into constituents” can be understood in different ways, but it seems clear enough that in §108 Russell has a specifically mereological sense of analysis in mind. This is strongly suggested by what he writes immediately after the passage we have just quoted: talk of definition in the sense of analysis into constituents overlooks “the fact that wholes are not, as a rule, determinate when their constituents are given, but are themselves new entities (which may be in some sense simple), defined in the mathematical sense, by certain relations to their constituents”. The point of the section is thus plausibly construed as the claim that the specifically mereological sense of analysis – and together with it, the Moorean notion of proposition202 – have no application in mathematics. One of the more pressing problems with philosophical definitions thus understood is that it is very difficult, both philosophically and mathematically, to make sense of the notion of “philosophical insight” or intuition that forms its semantic and epistemic basis. Kant had of course succeeded in assigning a tolerably clear content to “intuition”; at any rate, he had succeeded in giving it content that does not reduce to zero as soon as one begins to ask serious questions about it. But Kantian intuition is not equivalent with “philosophical insight”, or even “insight”, although this is what Kant’s term Anss-

202 See above, section 4.4.2. 472 Ch. 4 Logic as the Universal Science I chauung may suggest. Moreover, it would have been an embarrassment to Russell, to say the least, had he been forced recognize that intuition in anything like Kant’s sense does have an indispensable role to play in mathematical theories; after all, the philosophical point behind Russell’s variant of logicism was precisely to dispense with Kant’s theory of mathematics. Setting Kant aside, it is natural to suggest Russell does invoke intuition, after all, even if not under that very name. For he does speak about “acquaintance” and writes in the Preface to the Principles, that “[t]he discussion of indefinables [...] is the endeavour to see clearly, and to make others see clearly, the entities concerned, in order that the mind may have that kind of acquaintance with them which it has with redness or the taste of the pineapple” (1903a, p. xv; emphasis added). If “acquaintance” is used in the way Russell indicates here, however, it has no more content than “intuition” or “insight”. Russell himself acknowledges this, as he continues: “Where, as in the present case [sc. philosophical logic], the indefinables are obtained primarily as the necessary residue in a process of analysis, it is often easier to know that there must be such entities than actually to perceive them; there is a process analogous to that which resulted in the discovery of Neptune, with the difference that the final stage – the search with a mental telescope for the entity which has been inferred – is often the most difficult part of the undertaking.” Talk of search with a mental telescope can only be metaphorical, and if it is not, the content it is given tends to undermine the analogy with astronomy.

4.6.5 What Did Russell Really Say About Mathematical Theories

These philosophical difficulties with insight/intuition become pressing when the idea of philosophical definition is applied in mathematics. J. Alberto Coffa has provided a succinct formulation of the conceptual situation at which philosophers of mathematics, Russell, among them, had arrived by the turn of the 19th and 20th centuries: Ch. 4. Logic as the Universal Science I 473

By the end of the nineteenth century, the only reason anyone could possibly have for saying that the ultimate distinction between Euclidean and hyperbolic distance is given by acquaintance was the weight of a dead philosophy. One might insist that once we understand these notions, we have become acquainted with the concepts in question. But this could be viewed only as a linguistic ploy, designed to obscure the fact that acquaintance could not be asked to play in geometry the role that atomism had assigned to it. Circa 1900 it was no longer possible to suppose that it plays the specific semantic explanatory role in geometry that it was supposed to play in the atomist picture of knowledge, that is, that the construction of geometric theory would start with acquaintance, then proceed to a construction of claims, and perhaps conclude with the testing of these claims.203

Coffa (1992, p. 132fn. 18) and Shapiro (1997, p. 156) cite the following passage from Russell’s paper, Non-Euclidean Geometry, published in 1904, as evidence for his continuing adherence to semantic atomism, i.e. the view that acquaintance is what enables us to assign meanings to geometric primitives: “[I]t is, undoubtedly, by analysis of perceived objects that we obtain acquaintance with what is meant by a straight line in actual space” (Russell 1904b, p. 483). In fact, however, the text from which this passage is taken does

203 Coffa (1991, pp. 132-3). By “atomism” and “atomist picture of knowledge” Coffa means the following view: “if a sentence S is to convey information [...], then its grammatical units must have a meaning before they join their partners in S. Recognition of the meaning of the constituent phrases must be independent of and prior to the acceptance of the statement in question” (id., p. 131). Accepting this as a premise, one could then argue that since it is “evident” that the axioms of a geometry convey information – “express propositions”, as philosophers used to say – their semantically significant parts must possess pre-theoretic or pre-axiomatic meanings; this is how Frege and Russell reasoned, according to Coffa. Alterna- tively, given the atomist premise, one could argue that since geometric primitives evidently do not have independent meaning, the so-called axioms cannot convey (ordinary) information or express propositions; according to Coffa (id., pp 133-4), this was Poincaré’s line of thought. It implies geometric conventionalism, or the view that the axioms of a geometry do not convey ordinary information but express complicated conventions. 474 Ch. 4 Logic as the Universal Science I not support the conclusion that Coffa and Shapiro draw from it; considering the context of the quotation, we can see that the opposite is the case. What Russell is discussing here are the opinions expressed by Hugh MacColl. Russell argues that there is “a rather fundamental difference” between MacColl and himself, when it comes to the meanings of geometric primitives. As Russell reads him, MacColl holds the view that the concepts line, straight and point are simples, because their meaning is given through analysis of perceived objects. Russell says he is prepared to accept that there is a sense in which this is correct, for it is “by analysis of perceived objects” that we learn the meanings of the relevant terms, which is what enables us to “recognize as such the approximations to straightness which occur in nature” (ibid.) He emphatically denies, however, that this sense is relevant to pure mathematics (ibid.) Thus, far from accepting the view that Coffa and Shapiro attribute to him, he explicitly rejects it, explaining that it does not help us to determine the properties and relations of straightness. Drawing a distinction between pure geometry and geometry as the science of actual space, he argues that in pure mathematics “we do not consider actual objects existing in the actual world, but hypothetical objects endowed by definition with certain properties” (id., pp. 483-4). This leads to the following comment on the previously discussed notion of straightness:

In geometry, as a branch of pure geometry, we do not begin by inquiring into the nature of straight lines in actual space; we begin, instead, by considering hypothetical objects having properties suggested by actual straight lines, but not necessarily belonging to them. We consider many different sets of such properties, each set giving us a different kind of space. We prove, with each set of properties, that the various properties are compatible inter se, by actually constructing an assemblage of entities possessing these properties. (id., p. 484)

Note how this passage goes straight against the picture of geometrical knowledge that Coffa attributes to Russell. Pace Coffa, Russell did not think – or no longer thought – that “the construction of geometric theory would start with acquaintance”; since the entities studied in Ch. 4. Logic as the Universal Science I 475 pure mathematics are “hypothetical”, intuition or acquaintance can have no semantic explanatory role. Thus, the characterization that Shapiro applied to Hilbert applies to Russell as well: since some of the properties that we assign to these hypothetical entities may be suggested by suitable actual entities, like the approximations to straight line that we encounter in our experience, the role of ‘acquaintance’ must be limited to “motivation and heuristic”. Thus, Russell holds that the problems pertaining to “intuition” or “acquaintance” as potential sources of semantic information – and epistemic justification, we might add, though this question is barely touched upon by Russell in the Principles – are thus completely irrelevant in pure mathematics. It follows, also, that it is not important for our purposes to reach a conclusion as to how the Russell of the Prin- ciples construed the distinction between philosophical and mathematical definitions. Whether or not philosophical definitions are worth pursuing in some other domain, they play no role in pure mathematics, which is what matters for Russell. We must conclude then that the traditional conception of axiomatics, or the theory of philosophical definitions, cannot be used to throw light on the nature of pure mathematics, and is at least to that extent useless. By the time of the Principles, Russell was thus clearly accepting (a version of) the abstract conception of mathematics.

4.6.6 Russell and Abstract Axiomatics

Like other mathematicians of the time, Russell was led to the abstract conception of axiomatics primarily because he needed an account of what was happening in geometry.204 If the so-called axioms of a geometry do not receive content and truth-value by being about a subject-matter, how are they to be understood? If, in particular, the entities given through “spatial intuition” are at best irrelevant and at

204 This connection is stated by Russell himself in the Introduction to the second edition of the Principles; see Russell (1903a, p. vii). 476 Ch. 4 Logic as the Universal Science I worst non-existent, what kind of content do the axioms possess? We have already seen how Russell explained this to MacColl. Looking for a more precise characterisation, we come to see that Russell’s explanation in fact has two sides to it, corresponding closely to Bernays’ description of “abstract axiomatics”, which was cited above. Firstly, Russell explains that the assertions made in pure mathematics are hypothetical in nature. A pure geometer, that is to say, does not assert his theorems, but asserts a conditional A1...An T, where A1...An are the relevant axioms and T is a theorem that can be derived logically from the axioms. Geometry is thus primarily concerned with theorem-proving.205 In §8 of the Principles Russell explains that since the pure geometer asserts only implications, two mutually incompatible theories like the Euclidean and non-Euclidean geometries are “equally true”. This is a somewhat unfortunate formulation of the idea that, from the standpoint of pure mathematics, these theories are equally legitimate, since what matters is not truth (or truth simpliciter), but logical consequence, or proof.206 This formulation, however, implies no actual confusion on Russell’s part. The second part of Russell’s account has to do with the ‘semantics’ of the system. He argues – again, in essential agreement with Bernays’ description of abstract axiomatics – that even if a geometry as a branch of pure mathematics lacks intuitive subject-matter, it nevertheless can be said to be about something. §8 provides a prelimi-

205 For a very clear formulation of the basic idea, see the Principles, §5; see also §§352 and 353, where a more elaborate discussion of Russell’s version of ‘if-then -ism’ is to be found. What is significant about the philosophical motivation of this conception is that it “represents correctly the present usage of mathematicians”; another point of some importance is that Russell’s definition of mathematics “will produce, especially upon Kantian philosophers, an appearance of wilful misuse of words” (§352). 206 Some of the clarifications that Hilbert gave to Frege were similarly confused, as when he explained that as soon as one has laid down a set of axioms and shown that they do not contradict one another, then the axioms are true and the things defined by the axioms exist (Frege 1980, p. 39). Ch. 4. Logic as the Universal Science I 477 nary formulation of this idea. Russell argues there that what matters mathematically are the specific kinds of relations that obtain between entities belonging to certain classes, and not the entities themselves, or their classes.207 Since the actual terms of relations are irrelevant, a theory belonging to pure mathematics is said by Russell to be about a whole class of classes, namely all classes of entities having relations of the specified kinds. For instance, in §412 Russell explains that the axioms of finite arithmetic define a class of classes called progression, and that finite arithmetic applies to any of these classes, i.e. to any individual progression. Although Russell’ terminology is slightly eccentric, and his explanations somewhat cumbrous, the basic point is clear enough. It is tempting to formulate this by saying that, according to Russell, a branch of pure mathematics is about (a class of) relational structures; for instance, that Euclidean geometry as a theory of pure mathematics is about Euclidean spaces, and a Euclidean space is a class whose members are points together with relations between them. Ostensibly, this is exactly what Russell’s explanation amounts to.208 Russell’s construction is decidedly not model-theoretic in the full sense of the term. In addition to the fact that he lacks a notion of relational structure as a certain kind of set-theoretic entity, there are

207 In §8 of the Principles, Russell gives an example to illustrate the pure mathematician’s procedure. A statement like “Socrates is a man” does not belong to pure mathematics, or logic, because it contains non-logical constants. Substituting a variable for “Socrates” yields “x is a man”, and adding a suitable assumption yields “‘x is a Greek’ implies ‘x is a man’”, which holds whatever x may be. However, neither ‘Greek’ nor ‘man’ are logical constants. Therefore, they must be replaced by variables, and the eventual result will be “if a and b are classes, and a is contained in b, then ‘x is an a’ implies ‘x is a b’, whatever x may be”. This, Russell says, is a proposition of pure mathematics and is characterized by the fact that it gives rise to indefinitely many “chains of deductions”, depending on the values assigned to the variables. 208 As Hodges (1986, pp. 139-40) points out. 478 Ch. 4 Logic as the Universal Science I two other points that stand out. First, Russell does not have an explicit notion of “truth in a structure” or “true in a model” or “axioms satisfied by a model”, etc. But he does have the notion of a set of propositions defining a class of entities and any other class with the same relations holding in it, together with the further notion of a class satisfying the definition (see §122 of the Principles, and section 4.5.4 above). Since a single proposition can be said to provide a partial definition of a class, the relation specified by the proposition is in Russell’s terminology internal (to the class being defined; see §412 of the Principles). It is worth pointing out that Russell’s talk of definitions here is deliberate. He thinks that axioms should be true; hence the ‘axioms’ of a pure geometry are no more than “so-called axioms” (ibid.), and pure geometry, strictly speaking, does not “admit genuine axioms” (ibid.) Russell’s terminology thus conforms with Hilbert’s practice – which, as we have seen, was severely criticised by Frege. Second, Russell has quantified variables instead of schematic letters. A partial definition of a space might be a proposition like

(*) If a and b are any two entities, there’s a relation, R, holding between a and b.

R, a and b are of course subject to further conditions, and new entities may be introduced into the class by stating further conditions, which together yield a definition of a space. The definition, then, consists of propositions like (*), which, if written out in full, are quantified propositions about any entities; and what a geometer establishes is a hypothetical proposition to the effect that to entities of which a certain condition holds, certain other condition applies as well. As far as Russell is concerned, this picture of pure mathematics is complicated by the fact that his conception of arithmetic was logicist, rather than structuralist or if-then-ist. Russell, that is, did not hold the view that “the axioms or pseudo-axioms of finite arithmetic define a class of classes, called progressions” is all there is to the ontology of Ch. 4. Logic as the Universal Science I 479 natural numbers. As Gregory Landini (1998, p. 36) has pointed out, his view was rather that the Frege/Russell cardinals result from conceptual analysis of “natural number”. This, though, does nothing to change Russell’s recognition that there are other (“non-standard”) classes satisfying Peano’s postulates. I conclude, then, that Russell’s conception of the content of mathematical theories simply has to involve conceptual resources that are essentially identical to those that Hilbert had applied in his Grund- lagen der Geometrie. The abstract conception of theories allows for several implementations, but insofar as commitment to that conception implies a generic acceptance of model-theoretic conceptualizations, then Russell clearly accepted such conceptualizations.

4.7. Conclusion

In this chapter I have examined the universalist conception of logic from the standpoint of the van Heijenoort interpretation. We saw how, on this interpretation, the gist of the conception is the view that logic is universal in the sense of encompassing all thought (all principles underlying correct reasoning, etc.; the formulation varies). Given this, it is argued, the universalist has no room for an “external perspective” and is, for that reason, committed to rejecting metaperspective and metatheory. This in turn means that semantic concepts and semantic theorizing must be written off as illegitimate. A detailed case was then developed against this interpretation, starting from the observation that most of the concepts that the advocates of the van Heijenoort interpretation use are in fact ambiguous. Thus the term “logic” can refer to correct principles of reasoning (logic as science), or else to a calculus providing an explicit formulation of such principles. Since universalist logicians typically have use for both of the senses, the mere fact that they consider logic to be universal does not suffice to rule out a calculus for logic. There is thus room for meta- perspective as well, provided its function is correctly understood: (i) metatheory is about a calculus of logic; (ii) metatheory is about se- 480 Ch. 4 Logic as the Universal Science I mantic explanation rather than epistemic justification. The application of the van Heijenoort interpretation to Russell’s case was then examined in detail. And here it was found that much of what he has to say about logical theory follows straightforwardly from his view that the subject-matter of logic consists of propositions in the ontological sense (the notion of Russellian proposition). Thus he has ostensibly very little use for semantic argumentation. This simple picture gets complicated, however, as soon as the ambiguities mentioned above are taken into account. Further complications are due to the fact that a number of concepts can be fitted into Russell’s logical framework in spite of their ostensibly semantic character. For example, it is natural to think of truth as a property that belongs to representations (linguistic or otherwise) in virtue of their standing in a suitable relation to non- representational reality. The early Russell, however, rejects this correspondence intuition; for him, truth is a metaphysical notion, not semantic. Another illustration of this phenomenon is provided by the notion of interpretation. It is natural to think of interpretation as something that has to do with meanings. This applies even to the model- theoretic approach, which typically relies on a very thin conception of meaning. Russell, by contrast, has officially no room for such concepts (as we saw, the details are less clear on this point); nevertheless, he does have a notion of interpretation, namely the notion of interpretation by replacement, whose function is analogous to that of semantic interpretation. In this chapter my discussion of Russell’s version of the universalist conception of logic has been largely negative; I have tried to show what is not involved in it, although some positive elements have emerged as well. In spite of being critical of the van Heijenoort interpretation, I have said on more than one occasion that there is no denying that Russell’s conception of logic is universalist in some appropriate sense of that term. What that appropriate sense is remains to be determined. This task will be taken up in Chapter 5. Chapter 5 Logic as the Universal Science II: Logic as a Synthetic Apriori Science

5.0 Introduction

In Chapter 4 the universalist conception of logic was examined from the standpoint of the van Heijenoort interpretation. Several reasons were found why this interpretation cannot be regarded as adequate. To see what is really involved in Russell’s version of the universalist conception, we should examine what he means when he argues that logic is synthetic and apriori. In the present context, both of these notions carry senses that are distinctly anti-Kantian. My suggestion, therefore, is that we can gain a better understanding of Russell’s views on logic, if we consider them against the background of what Kant had to say about formal logic. In section 3.6 I examined Russell’s criticisms of Kant’s account of how propositions that are synthetic can nevertheless be known apriori. Russell’s arguments, I maintained, were directed against the model of explanation assumed in Kant’s explanation. Kant, it is argued, sought to derive a number of properties of propositions from features characterizing our cognition of these propositions; this was referred to as the relatvized model of the apriori or the r-model. One of the arguments against the r-model was that the ground it proposes for the synthetic apriori is too weak to sustain the characteristics that philosophical tradition has attached to apriority. Russell’s own view was that the source of the synthetic apriori in mathematics is to be found in logic. Since logic is apriori, an acceptable account of its propositions is constrained by conditions revealed through an examination of the r-model. On the face of it, this implies that the propositions of logic must be true, universal and necessary. Each of these characteristics, moreover, must be genuine rather than proxy as in Kant’s theory, according to Russell. The propositions of logic are also synthetic. In the present context, this amounts to the requirement that they must have content. Ap- 482 Ch. 5 Logic as the Universal Science II plied to the early Russell’s conception of logic, the notion of having content implies no less than four distinct claims about the nature of logic, all of them related to what Kant had to say about formal logic:

$ Logic does not abstract entirely from the relation of thought to objects. $ Logic is about the world. $ There is no “deep” distinction between the form and content of propositions; in particular, the notion of form does not do independent, explanatory work, when it comes to the propositions of logic. $ Logic is capable of providing the foundation for mathematics.

Each of these four theses goes directly against Kant’s conception of formal logic. For him, the key to thinking about this kind of logic is the principle that I shall call the analyticity-constraint. This condition is a complex one, but the first approximation would be that, for Kant, it implies lack of content in all of the four senses distinguished above. These two perspectives on logic – corresponding to the claim that it is synthetic and the claim that it is apriori – can be brought together by considering how the constraints on apriority and the idea that logic has content together constitute the essential ingredients of the early Russell’s universalist conception of logic. It is precisely these characteristics, furthermore, that can be shown to figure in Kant’s understanding of what formal logic is not. The question of demarcation is often regarded as one of the key questions in the philosophy of logic. Whether or not this is correct, it is clearly an important question for a philosophical project like Rus- sell’s logicism. In the absence of a non-arbitrary criterion of logicality, logicism would not have the consequences Russell took it to have; in particular, he would not be in a position to argue that Kant was wrong about the nature of mathematics. Since Russell was an advocate of the universalist conception of logic, one might think that his criterion of logicality would have Ch. 5 Logic as the Universal Science II 483 something to do with the normative character of logic, i.e., with the fact that logic in some sense constitutes the principles of correct reasoning. It turns out, however, that this idea – present, for instance, in Kant and Frege – plays no important role in Russell’s thinking about logic. Instead, his version of the universalist conception operates with a purely descriptive, as opposed to a normative, notion of generality. I will address two key issues relating to this notion. Firstly, I will examine the context in which it emerges; this is constituted by what I will call the Bolzanian account of logic. This revolves around the concept of logical constant. Secondly, I will examine what Russell has to say about the propositions of logic and, in particular, about logical inference. Here one can see the Russellian version of the universalist conception at work, and it is here that its consequences are clearly visible.

5.1. Hylton on Russell’s Commitment to the Universalist Conception of Logic

According to Peter Hylton, there is a fairly direct connection between Russell’s commitment to the universalist conception of logic and his anti-Kantianism. In this section I will consider Hylton’s interpretation, arguing that it cannot be upheld. After this, I will explore what I take to be the correct view of the matter. Recall Hylton’s view, discussed in sections 3.5.1 and 3.5.2, that among Russell’s anti-idealist arguments there is the largely implicit line of thought that mathematics, understood as in logicism, constitutes a particularly clear counterexample to the idealist doctrine that ordinary thought and knowledge are at best something relative (non- absolute). Kant’s view, roughly, was that human thought and knowledge is confined to appearances, while the thing-in-itself is necessarily beyond our grasp. Post-Kantian idealists argued in turn that while metaphysics, with its untainted categories, could deliver the truth about reality as it really is, non-metaphysical thought could give us 484 Ch. 5 Logic as the Universal Science II only appearances.1 Russell thought, according to Hylton, that the logicist reconstruction of mathematical knowledge constitutes a clear counter-example to these idealist claims and provides a strong argument for logicism and against idealism. For idealism does not allow mathematics to be true in the non-corrupt sense accepted by everyone save for idealist metaphysicians, while logicism permits mathematics to be true precisely in this natural and straightforward sense. If, now, mathematics is reducible to logic, logic, too, must possess this property as well. This part of Hylton’s reconstruction is essentially correct; Russell did defend the absolute character of ordinary truth-ascriptions (as in mathematics) as against what he took to be idealist aberrations; mathematics gives us truth, and here truth is not what idealists take it to be, but the real, absolute notion. This view of Russell’s is important, because it figures prominently in his criticism of Kant’s explanation of the apriori.2 This, however, is not the direction in which Hylton develops the argument. His primary aim is to show that the philosophical use to which the early Russell put logicism committed him to the universalist conception of logic as this is understood on the van Heijenoort interpretation. More specifically, Hylton argues that Russell was bound to think about logic along the universalist lines, rather than model-theoretically, because only the universalist conception is compatible with the role that logic plays in his anti-Kantian – more generally, anti-idealist – argumentation (see, e.g. Hylton 1990b, pp. 196-7). Un- fortunately, Hylton is less than fully clear about the details of the argument he intends to attribute to Russell. What is clear is that the focus is on the notion of truth:

1 As was pointed out in section 3.5.1, this generalization neglects whatever differences there may have been among “post-Kantian idealists”. For instance, Bradley was an important exception, as he thought that no matter what categories we may use, they cannot give us more than Appearances. 2 See section 5.10. Ch. 5 Logic as the Universal Science II 485

Given that Russell’s use of logicism as part of an argument against Kant and the idealists is as I have described it, what does this imply about Russell’s conception of logic? To play the philosophical role that Russell had in mind, logic must, above all, be true. Its truth must be absolute, unconditioned and unrestricted. These features may appear to be uncontroversial, even trivial, but in fact they mark a crucial difference between Russell’s conception of logic and what I have called the model-theoretic conception. (id., p. 205)

Reading this passage literally, we must conclude that, according to Hylton, there are no less than three characteristics that the concept of truth must have for Russell: the truths of logic must be absolute, unconditional and unrestricted. It is not immediately clear how these terms are intended to be understood, but the following figure (see figure 5.1) provides one suggestion as to how the relevant notions might unfold (UCL is shorthand for “universalist conception of logic and TI for “transcendental idealism”: 486 Ch. 5 Logic as the Universal Science II

Russell’s UCL Kant’s TI model- theory

truth-value absolute dependent upon truth-in- ascriptions sensible and an- conceptual syn- interpreta- thesis tion

subject- the objects of the objects of matter de- knowledge are knowledge ? pendence independent of partly consti- facts about our tuted by sensible cognition and conceptual synthesis

Universality one universal do- confined to the several (pos- main comprising objects of possi- sibly irre- “all objects” ble experience ducible) do- (spatiotemporal mains for realm) quantifiers

there is one logic several logi- (one set of princi- cal languages ples for correct thought)

Figure 5.1 Hylton on Russell, Kant and model theory on the concept of truth

Here “truth-value ascriptions” is an explication of Hylton’s “absolute”, “subject-matter dependence” corresponds to his “unconditional”, and “universality” is intended to capture Hylton’s “unrestricted”. In what follows I shall comment on each of these three features. (1) Truth-value ascriptions. According to Kant, the objective content of a judgment depends upon conceptual and sensible synthesis. The rela- Ch. 5 Logic as the Universal Science II 487 tion to the sensible manifold is one that first emerges through synthesis and is that which puts judgments into contact with objects. With- out this relation, judgments would have no content.3 And without object-relatedness and content, there would be no truth, either.4 Ac- cording to Kant, synthesis is something subjective; it is “the act of putting different representations together, and of grasping what is manifold in them in one act of knowledge” (A77/B103)). The possibility of objective judgments and, with it, of truth-value ascriptions is therefore conditional upon an activity effected from the side of he knowing subject. In model theory, too, truth-value ascriptions are dependent upon something else: the sentences of a formal language possess their truth-value only relative to an interpretation. A sentence is in the first instance no more than a string of symbols. In order for the sentence to receive the value “true” or “false”, its semantically significant parts must be assigned interpretations. The upshot is that the model-theoretician’s truth-bearers are not true (or false) simpliciter but only relative to or in an interpretation. Russell, by contrast, holds that truth-bearers are propositions in his sense. And this means that they possess their truth-values independently of language, interpretation or

3 As we saw in chapter 2, if Kant’s analysis of their content is correct, this applies even to mathematical judgments. For Kant’s own explanation of synthesis, see A77-80/B102-105. 4 In the absence of object-relatedness, all that is left are cognitions standing in logical relations to one another. Since truth consists in “the agreement of [a cognition] with its object” (A58/B83), a necessary condition for truth is lacking, insofar as object-relatedness is not taken into account. Kant argues that formal logic, which is the part of logic that abstracts from the content of cognition and deals exclusively with its form is capable of yielding rules that together define a purely negative criterion for truth, namely absence of contradiction. But such rules “concern only the form of truth” (A59/B80), i.e., the different logical forms that judgments, which are the vehicles of truth, may take; such rules in themselves are not sufficient as a criterion of truth, because a cognition that is logically possible may still contradict its object, that is, may not be compatible with the conditions of synthesis or objecthood. 488 Ch. 5 Logic as the Universal Science II acts of synthesis.5 In our terminology, Russellian truth-value ascriptions are absolute, rather than relative. (2) Subject-matter dependence. By this term I mean the view, typical of idealism, that the subject-matter of judgments is partly dependent upon something subjective. In Kant’s transcendental idealism the claim is that judgments are dependent upon the knowing subject not only in the sense of truth-value ascriptions but also in the sense that, for example, space and time qua conditions of synthesis and objective knowledge are transcendentally ideal. This means, according to Kant, that space and time are not attributes of things-in-themselves, but something that we in some sense have put into the objects of cognition. Kant’s view is often explained to mean that objects are, or can be described as, spatial and temporal “only insofar as” they are, or are regarded as, potential objects of human cognition. Russell’s realism, by contrast, is characterized by an unconditional rejection of all such idealist claims; the objects of knowledge – Russellian propositions and their constituents – are what they are and do not depend upon synthesis for their existence or character. The possibility of cognition or the fact that we have knowledge is therefore entirely irrelevant, as far as the objects of knowledge are concerned. In contrast to Russell and Kant, model theory is silent about such metaphysical issues. That is, model theory, considered in itself, is entirely neutral on the metaphysical status of the various ingredients that define the notion of a model; in particular, model-theory is non- committal on whether the condition of a sentence’s being true should be understood in the manner of “realism” or “idealism” or “anti- realism” or some other way. This is why I have inserted a question mark in the relevant box describing the commitments of model theory. (3) universality. Considering the first two rows of the above table, it seems clear enough that no relevant difference emerges between Rus- sell’s universalist conception of logic and the model-theoretic concep-

5 Cf. Hylton (1990b, p. 205). Ch. 5 Logic as the Universal Science II 489 tion with respect to Kant’s philosophy. That truth-value ascriptions are mentioned at all seems to be due to a failure to distinguish between the idea that truth-value ascriptions are relative and the idea that truths themselves should be dependent upon something subjective. In model-theory, truth-value ascriptions have to do with the expression of truth, and even if that is a relative matter, the relativity is metaphysically innocent. For when Russell takes Kant to task for his transcendental idealist model of the synthetic apriori, the root complaint is that this model undermines the relevant judgments’ claim to knowledge; this complaint is about subject-matter dependence, not about truth-value ascriptions. I conclude that a difference in their respective treatments of truth- value ascriptions cannot distinguish the universalist and the model- theoretic conceptions of logic in any way that would be relevant to the dispute between Russell and transcendental idealism. Instead, the gist of the dispute has to do with subject-matter dependence (the second row in the table). It has just been pointed out, however, that model theory has no implications whatsoever for this metaphysical issue, nor, therefore, for the dispute. If these conclusions are correct, it follows that universality is the only feature on which the universalist and the model-theoretic conceptions might turn out to disagree in ways that are potentially relevant to Russell’s criticism of transcendental idealism. Even here, however, it is doubtful whether Hylton’s line of thought can be upheld. To begin with, it should be noted – as the lowest row of the table indicates – that universality may mean two quite different things. Firstly, it may mean the idea that logic applies to everything, i.e., that the generality that is characteristic of logic is unrestricted in the strongest sense: the variables occurring in the propositions of logic range over everything, and here “everything” must be taken literally, subject to no contextual restrictions.6 Sec- ondly, universality may refer to the idea that there is one logic within the scope of which all (deductive) reasoning falls. The question is:

6 See section 4.5.6.2 for a brief elaboration of this idea. 490 Ch. 5 Logic as the Universal Science II what do the two senses imply for Russell’s anti-Kantian argument? The first sense of universality appears to have little relevance to the issues that occupied Kant and Russell. The disagreement between the two has to do with whether our concepts must be regarded as being intrinsically restricted to the objects of possible experience, as Kant thought, or whether the introduction of mind is at all needed in an explanation of the synthetic apriori. Again, we can safely conclude that model theory per se has nothing to contribute to these questions. Given a formal language, its interpretation includes the specification of a domain of objects from which appropriate interpretations are chosen. Since the domain can be chosen whichever way one wants to and varied at will, the fact that one operates with restricted domains imposes no intrinsic limitation on the content of one’s concepts. On the other hand, there is, on the face of it, nothing to prevent one from stipulating that the domain is to include absolutely everything – and if the arguments considered in section 4.5.6.2 are correct, the notion of “unrestrictedly everything” is in fact presupposed, when we quan- tify over less than absolutely everything. Whichever way one decides to operate with the notion of domain, everything that the universalist logician can do can be accomplished in model theory as well. It turns out that the table given above is misleading; the first sense of universality, namely generality, is not an independent issue, but has to do with truth-value ascriptions: sentences have truth-values only relative to an interpretation, and interpretation includes a specification of a domain. Since we have already discussed truth-value ascriptions, this sense of universality can be put aside. We must next consider the second sense of universality. It is here that we find the gist of Hylton’s interpretative argument. The following quotation shows how he perceives the connection between the universalist conception of logic and the philosophical use to which the early Russell wanted to put logicism:

Given such a conception of logic there can be no external perspective. Any reasoning will, simply in virtue of being reasoning, fall within logic; any proposition that we might wish to advance is subject to the rules of Ch. 5 Logic as the Universal Science II 491

logic. This is perhaps a natural, if naive, way of thinking about logic. In Russell’s case, however, we can say more than this to explain why he should have held such a conception. Given the philosophical use that Russell wishes to make of logicism, no other conception is available to him. If logic is to be unconditionally and unrestrictedly true, in the sense that Russell must require it to be, then it must be universally applicable. This in turn implies that statements about logic must themselves fall within the scope of logic, so the notion of a meta-theoretical perspective falls away. If this were not so, if logic were thought of as set up within a more inclusive metalanguage, then by the standards which Russell and the idealists share, it would appear that logic is not absolutely and unconditionally true. Logic, on this modern picture, is not unrestricted, for it is set up in a more inclusive language which must fall outside its scope. Nor can the truth of logic, conceived of in this way, be thought of as absolute and unconditioned, for it is dependent upon the metalanguage within which it is set up. (1990b, pp. 206-7; emphasis in the original)

There are in fact two interpretative arguments in this passage. The last sentence is about truth-value ascription in the sense discussed above; what is “set up in a metalanguage” is the truth of logic, and this presumably means giving a truth-definition for a formal language. What is dependent upon metalanguage is thus the expression of truth – assignment of truth-conditions to a syntactic object. I have already concluded, however, that this kind of dependence is entirely harmless, insofar as Russell’s criticism of transcendental idealism is concerned. Hylton’s first argument is more important. The claim is that, for Russell, logic must be universally applicable in the sense that that each and every piece of (deductive) reasoning must fall within the scope of one logic. According to Hylton, this view contrasts with the more modern view that reasoning in metalanguage falls outside the scope of the object language. Hylton argues next that on the conception of absoluteness that Russell shares with 19th century post-Kantian idealists, the model- theoretic conception must be taken to represent a version of non- absolute, or less than all-inclusive, truth. That is, a model-theoretic logic is not absolutely true, because there are reasonings, like reason- 492 Ch. 5 Logic as the Universal Science II ing in meta-language, that fall outside its scope.7 Hylton elaborates on this claim by suggesting (id., pp. 208-9) that the contrast between the two conceptions of logic with respect to the second sense of universality has analogues in certain arguments that post-Kantian idealists directed against transcendental idealism. These arguments were intended to undermine Kant’s introduction of the thing-in-itself, or the idea that there is something outside the scope of our categories and hence of our thoughts. Some such argument is at the heart of non-Kantian or non-transcendental idealism, which often defined its own position by explaining how it differed from Kant’s. Very briefly, the difficulty that idealists raised for Kant was this: if the forms of sensibility and the concepts of understanding truly mark the limits of human cognition, and if thing-in-itself is supposed to contrast with the appearances that are delineated by these forms and concepts, how can we even have thoughts about this something falling outside our cognition? Post-Kantian idealists, in other words, criticized Kant for not being sufficiently serious about his own notion of category.8 Hylton’s suggests that Russell might have accepted an analogous argument against the model-theoretic conception. Hylton’s interpretative argument thus promises a single standpoint from which Russell’s views on logic can be related to both Kant’s transcendental idealism and the model-theoretic conception of logic. According to Hylton, then, there is, from Russell’s perspective, an important analogy between Kantian categories and logic. The first member of the analogy can be spelled out as follows (cf. Hylton 1990b, p. 209). If the categories that Kant has identified really are what he says they are, they must apply to everything: there is nothing – no thought – that can be regarded as not being subject to these concepts; and this means, among other things, that “they must apply to the critical philosophy itself, and thereby to the very statement of the

7 The term “true” is not very apt here, as reasoning is not true or false. Since, however, there is a connection between truth and validity, this terminological issue is insignificant. 8 Here I follow Hylton’s formulation (1990b, p. 209). Ch. 5 Logic as the Universal Science II 493 categories themselves” (ibid.) The second member of the analogy is not given explicitly, but if there really is a similarity between the two cases, the following should do as a reconstruction; if the rules and principles that a logician has identified really are what he says they are, they must apply to everything: there is nothing – no thought – that can be regarded as not being subject to these rules and principles; and this means, among other things, that they must apply to logic itself, and thereby to the very statement of these rules and principles themselves. The point of the analogy is thus that both Kant’s understanding of categories and the model-theoretic logician’s talk of logic are self-defeating; if categories and logical principles apply wherever there is thought, there can be no thought to which they do not apply. Formulated in this way, the weaknesses of the analogue become quite evident. The basic point is that it the analogue ignores the distinction between logic as a science and logic as a calculus which was found to be relevant even on the universalist conception of logic. Accordingly, it fails to observe that “external perspective” can be understood in two very different ways. (Cf. here section 4.5.2.2). Examination of the van Heijenoort interpretation led to the following two distinctions:

External perspective

(a) meta-perspective in the minimal sense in which meta- perspective on X means a perspective which one assumes when one is discussing X or is thinking about it; (b) perspective on X that is outside X.

Logic

Putting these together yields four possible combinations: (a + c), (a + d), (b + c) and (b + d). Insofar as “external perspective on X” simply means meta-perspective in the minimal sense, i.e., the sense in which one adopts an external perspective on X when one discourses on or thinks about X, then it is quite clear that Russell had no qualms about it; as a working logician he needed not only the idea that logic is a universal science of correct principles of reasoning; he also needed a calculus in which reasonings can be reproduced, and in establishing such a calculus, one must adopt an external perspective in the minimal sense. Hence, the fact that Russell’s conception of logic was universalist gives no cogent reason to think that he should have found (a + c) in any way illegitimate; and there are good reasons to think that he did not. Hylton’s analogy, on the other hand, assumes the second sense of “external”, i.e., (b) rather than (a).9 However, to combine (b) with (d) to formulate an argument against the model-theoretic conception of logic (or any other conception) would be most unreasonable, for no one is likely to assume that the possibility of theorizing about the norms for all thought – more modestly, about the norms for all deductive reasoning – presupposes that one could step outside these norms, i.e., that one could act as if these norms were not in force. Now, it is not unthinkable that Russell should have endorsed unreasonable or otherwise poor arguments. It is nevertheless quite clear that insofar as he did have arguments against the possibility of an external perspective, these were directed against certain kinds of reasoning that assume (a + d), rather than (b + d). Recall, for instance, Russell’s argument that the independence of a particular choice of logical axioms cannot be proved in the standard way; in the course of an attempted demonstration one would have to assume a principle of

9 This is shown by the following quotation, in which Hylton discusses the first member of the analogy: “If the categories really are categories, then they must apply to everything. There is nothing that we can conceive of as being exempt from them, and no position from which we can think without employing them” (1990b, p. 209). Ch. 5 Logic as the Universal Science II 495 inference false, and this cannot be done, for if the principles underlying a particular piece of reasoning are false, the reasoning becomes fallacious and does not establish its intended conclusion. This argument is a particular case of what holds more generally: since X is supposed to be a fundamental principle for all deductive reasoning, there are certain things which cannot be done about X. Two salient examples would be (i) giving a deductive justification of X; (ii) drawing consequences from the assumption that X is not valid (Russell’s argument, discussed in section 4.5.4; the first example was discussed in section 4.5.2.2.4). That these particular kinds of reasoning are illegitimate does not show that there could not be any reasoning about the principles of correct reasoning (or about a calculus purporting to codify such principles). And I do not think there is any indication that Russell ever thought otherwise. We are thus left with the combination (b + c). This is the idea that, given a calculus for logic (or, in more modern terms, a language for logic), there will always be reasonings that cannot be formulated or expressed in that calculus (language). Here, however, we are no longer discussing logic as a science, but a particular formulation of this science. And here one can argue as follows. If it turns out that there are reasonings which are external in sense (b), that discovery is in itself no embarrassment to an advocate of a universalist conception of logic; it only shows that logic in the fundamental sense (logic as a science) cannot be captured by any one calculus for logic.10 One could argue that the model-theoretic conception will not do as a conception of logic for Russell precisely because it ignores the logic-as-science aspect.11 How much there is to this suggestion depends upon what is involved in the idea of logic as a science – the idea that logic delivers the correct principles of reasoning – and what relation the model-theoretic conception bears to it. For instance, it is easy to imagine Russell criticizing the model-theoretic conception for adopting a linguistic conception of the subject-matter of logic. Hyl-

10 Cf. here Landini (1998, p. 36). 11 This point is perhaps made in Hylton (1990b, p. 208). 496 Ch. 5 Logic as the Universal Science II ton’s analogy, however, does not address such issues. Thus we reach the following general conclusion. Whatever reasons Russell might have had for faulting the model-theoretic conception, these are not concerned with the possibility of metaperspective. I do not think, therefore, that Hylton’s analogy succeeds in illuminating the philosophical underpinnings of Russell’s universalist conception of logic. This conception can nevertheless be illuminated through a comparison with Kant. To this effect we should consider what is involved in Russell’s claim that logic is a synthetic apriori science, a claim that has a distinctly anti-Kantian content, which can be better grasped by relating it to Kant’s views.

5.2. Kant on Formal Logic

5.2.1 Preliminary Remarks

In section 3.6 I examined in detail Russell’s criticisms of Kant’s explanation of the synthetic apriori. Russell’s starting-point is the attribution to Kant of a relativized model of the apriori (r-model for short). The r-model seeks to explain the synthetic part of apriori knowledge by locating its ground in certain standing features of human cognition (exactly how this is supposed to take place is something that was discussed in detail). Since apriority has traditionally been taken to imply truth, necessity and universality, the r-model must explain each of these properties. One of Russell’s criticisms of the model was the indirect argument which does not question the intelligibility of the proposed explanans – he does expresses doubts about that, too, but that is another issue – but purports to undermine Kant’s explanatory strategy by showing that the putative transcendental foundation for the synthetic apriori is too weak to sustain the properties that accompany apriority. The logicist Russell argued that the source of the synthetic apriori in mathematics is to be found in logic. Since logic is apriori, any the- Ch. 5 Logic as the Universal Science II 497 ory of logic is constrained by conditions that can be identified with the help of the r-model. On the face of it, this yields the following three conditions: being apriori, the propositions of logic must be true, universal and necessary. Each of these characteristics, furthermore, must be shown to be genuine, rather than a proxy – that the r-model does not allow this is one of the arguments that Russell directs against it. There is one complication that must be added to this otherwise straightforward account. It is that Russell did not really believe the propositions of logic to be necessary in any deep sense. As will be seen, this feature is perfectly compatible with his use of the r-model and does nothing to undermine its explanatory utility. The propositions of logic must be synthetic as well, according to Russell. What this means is best brought out by considering the requirement that they must have content. And this in turn can be understood by relating it to what Kant had to say about the formal logic, for the content-character of logic goes directly against everything that he thought about the matter. These two perspectives on propositions of logic can be brought together by considering how the constraints on apriority – those that can be identified with the help of the r-model – and on syntheticity – the idea that logic has content – constitute the essential ingredients of the early Russell’s conception of logic. As we shall see, it is precisely these same characteristics that figure prominently in Kant’s understanding of what logic is not. Examination of Kant’s conception of formal logic leads to the question of logicality or demarcation: what is it that sets logic apart from what is not logic? For Kant the question was important because it enabled him to identify the core question of critical philosophy. Ac- cording to Kant, formal logic is grounded in that part of human cognition which is epistemically unproblematic and for which the critical question does not arise. To provide a criterion of demarcation for formal logic can therefore be used to identify the problematic part of human cognition, that for which the critical question does arise. In Russell’s case the question of demarcation or logicality is 498 Ch. 5 Logic as the Universal Science II equally important, though the reasons are rather different. Logicism, as Russell understood it, is a philosophical thesis. To argue that mathematics is reducible to logic is to endorse a substantive view on the nature of mathematical concepts and truths and inferences. It is a familiar point that it is not enough for a logicist, insofar as he is anti- Kantian, to present a reduction of this or that portion of mathematics to a discipline he calls logic and argue that this shows the Kantian conception to be wrong. For a defender of Kant can always rejoin by claiming that the discipline underlining the reduction is not really logic at all.12 In Russell’s case the question of demarcation arises in a very acute form, and the reason for this is to be found precisely in his view that logic is synthetic in the sense we are about to explore; for this means, among other things, that logic is epistemically fruitful in a quite substantive sense. Kant, by contrast, endorses a view on the nature of logic that is the exact opposite of this; for him, logic is first and foremost characterized by an analyticity-constraint, which implies that logic can have none of the positive properties that Russell attributes to it. And it must be said that the philosophical community has tended to follow Kant, rather than Russell, on this point.13 Kant’s own version of the analyticity-constraint is a relatively straightforward consequence of his extremely narrow conception of what counts as formal logic. No one whose understanding of logic is

12 For a recent discussion of this issue, see MacFarlane (2002). 13 The case of Poincaré offers a good illustration of the power of the analyticity- constraint. Poincaré was in fact so convinced of the correctness of the constraint that he tended to dismiss Russell’s logicism by claiming that it introduced no more than a new way of using the term “logic”. Mathematics, according to Poincaré, is rich with intuition, and if a portion of mathematics can be developed from some base theory, that only shows that the base theory involves synthetic apriori judgements and is, therefore, dependent upon intuition. For Poincaré the paradigmatic example of this dependence is complete induction or “inference by recurrence”, which, he felt, could not be reduced to purely logical reasoning; see section 3.1.5.4 for further discussion of this particular example. Ch. 5 Logic as the Universal Science II 499 conditional upon modern rather than traditional logic can endorse Kant’s specific reasons for the constraint. Nevertheless, the constraint itself need not be given up on that account, and one can continue to use it against conceptions of logic that are similar to the one that underlies logicism. Logicism, after all, is grounded in the assumption that logic is a source of substantive knowledge of the world (objects, etc.),14 and there will be plenty of scope between this view and the Kantian one for conceptions of logic that incorporate one version or other of the analyticity-constraint.

5.2.2 Formality and the Analyticity-Constraint in Kant

In this section I will explain the essentials of Kant’s conception of formal logic. This will be done by explaining in what sense formal logic is formal, according to Kant, and how this leads to the analyticity-constraint.15 This perspective is useful in that it leads directly to the question of logicality, and does so in a way that is easily connected with Russell’s views on the subject. A philosopher trying give a short and concise characterisation of

14 There are, of course, versions of logicism that do not include this assumption. For example, logical empiricists were attracted by logicism precisely because they accepted the analyticity-constraint (cf. section 1.2.3). The reducibility of mathematics to logic shows, according to them, that the two are epistemically on par; because logic does not yield substantive knowledge of the world (the analyticity-constraint), the same goes for mathematics, too. Another point that should be kept in mind is terminological (this point has already been made in section 1.2.4, but is worth repeating here), to wit, that one’s attitude towards the constraint cannot be read off from one’s use of such terms as “analytic” and “synthetic”. Frege, for example, rejected the analyticity-constraint but accepted that logic, and therefore arithmetic, is “analytic”. 15 The following discussion on formality relies heavily on recent work by John MacFarlane; see MacFarlane (2000), (2002). Another up-to-date discussion of Kant’s views on logic that I have also consulted is Wolff (1995; see in particular chapter 3). 500 Ch. 5 Logic as the Universal Science II logic might resort in part to some such description as , “logic is topic- neutral”, “logical rules and laws of independent of any particular subject-matter”, or “logic is concerned with form rather than content”. Statements like these can be developed in different directions to yield several distinct explications of logicality. Since it is common to understand these statements as giving expression to the intuition that logic is a formal discipline, the resulting explications can be dubbed explications of logical formality. MacFarlane (2000, Ch. 3) distinguishes between three such notions of logical formality. They are familiar not only from literature on Kant but also, and in particular, from discussions of how logic should be demarcated. They are not the only notions available, but, MacFarlane argues, they are “the main notions in play” (id., p. 50). They are as follows:16

x Formality-as-normativity: logic is formal in the sense that it constitutes norms for the use of concepts as such, i.e., norms to which any conceptual activity – asserting, inferring, judging, etc. – must be held accountable. x Formality-as-non-particularity: logical rules and laws are formal in the sense that they abstract from the particular identities of objects (from the thisness of individual things and from the suchness of properties). x Formality-as-non-substantiality: logic is formal in the sense that it abstracts entirely from the semantic content of concepts, i.e,. it considers thought in abstraction from its relation to the world.

These three senses are by no means mutually exclusive. When we turn to Kant’s views on logic, we shall see that all three are in fact relevant to how logic is understood in the context of transcendental idealism. In Kant’s mature philosophy, the most comprehensive description that can be given of logic is that it is the “science of the rules of the

16 The labels are mine, not MacFarlane’s. Ch. 5 Logic as the Universal Science II 501 understanding in general” (A52/B76); the contrast is with the rules that pertain to sensibility (aesthetic). Characterized in this way, logic is not one science but many. The most fundamental division is that between general logic and special logics. Of these, the former “contains the absolutely necessary rules of thought without which there can be no employment whatsoever of the understanding” (ibid.), whereas the latter contain “the rules of correct thinking as regards a certain kind of objects” (ibid.) Kant’s own innovation, transcendental logic, is one among the special logics. It constitutes the logic of metaphysics; even if the notion of object that is relevant to metaphysics is quite general – it is in fact the most general notion of object – transcendental logic still counts as a special rather than general logic. General logic is either applied or pure. Applied general logic considers the “empirical conditions under which our understanding is exercised” (A53/B77), and is thus a branch of empirical psychology. Pure general logic, by contrast, abstracts from all such contingent conditions and considers only those principles of the understanding that apply necessarily, i.e., apply if there is to be an employment of the understanding at all (see Kant’s explanation at A52/B76). Here “contingent” and “necessary” do not refer to contingent and necessary truth but to the idea that the rules in question are either conditional or absolute norms for thought. These two kinds of norms are operative in the distinction between special logics and general logics as well: since it is contingent that my current thought has the subject- matter that it does have (whatever it may be), the rules that pertain to that subject-matter apply to my thought only conditionally, that is, only insofar as my thought has that particular subject-matter. The rules of pure general logic, by contrast, are not in this way topic- sensitive: the rule, for instance, that if a thing has a property, it cannot at the same time not to have it, applies to my thought, no matter what it is that I am thinking about. Central to Kant’s characterization of logic is the issue of object- relatedness. Since objects are given to us only through sensibility, the faculty of understanding, considered independently of its relation to 502 Ch. 5 Logic as the Universal Science II sensibility, constitutes mere thought or thought as such. That it can be so considered means that there are rules for thought that apply independently of sensibility; hence, although Kant is usually quite insis- tent that thoughts wit hout intuition are empty, there nevertheless is a minimal sense of “thought” that is available independently of thought’s relation to sensibility. Once object-relatedness is taken into account, it will be found that there are not only rules whose source is sensibility – spatiality and temporality qua forms of objects – but also rules that concern the conceptual side of object-related thought; hence, in addition to general logic, there are also special, “objectual” logics. Kant’s understanding of logic is thus firmly embedded in his conception of the elements that can be distinguished in objective thought, i.e., the sort of thinking about objects that constitutes human cognition:

(*) Conditions for thought as such (pure general logic)

(*) Conceptual conditions for thought- with-objects: – conditions for pure thought of objects (transcendental logic)

(*) Sensible conditions for thought-with-objects: – spatiality and temporality as forms of objects (transcendental aesthetic)

Figure 5.2 The conditions of objective thought, according to Kant

When one describes the various elements that together constitute the conditions of objective thought, it is in many ways natural to resort to talk of form and content. Figure 5.2 suggests, however, that there is not just one but several notions of form that could be applied to the Kantian picture of human cognition. For instance, if one decides to Ch. 5 Logic as the Universal Science II 503 use the term “logical form” to refer to the conditions of thought as such (the realm of pure general logic), then, from this point of view, the conditions pertaining to thought-with-object belong with content rather than form. However, when one’s focus is on thought-with- objects, its conditions are naturally regarded as constituting yet another formal aspect of objective thought, though the sense that “formal” has here (the schematized logical forms identified in transcendental logic and the spatial and temporal forms identified in transcendental aesthetic) evidently differs from the sense that is relevant to thought as such. Turning now to a more detailed consideration of Kant’s pure general logic – the science which comes closest to our formal logic – the question arises what relation the three senses of formality that were distinguished above bear to it. As we have already seen, one of Kant’s characterizations of pure general logic is that it “contains the absolutely necessary rules of thought without which there can be no employment whatsoever of the understanding” (A52/B76). That the necessity to which Kant is referring in this passage is normative may not come out as clearly as it should, but the point is made very clearly by Kant in the introduction to the Jäsche Logic:

In logic we do not want to know how the understanding is and does think and how it has previously proceeded in thought, but rather how it ought to proceed in thought. Logic is to teach us the correct use of the understanding, i.e., that in which it agrees with itself. (Kant 1992, p.14 (529))

On the face of it, there is an evident problem facing the suggestion that formality in this sense, or formality-as-normativity, could be the fundamental sense in which logic is formal. For the “as-such” in “thought-as-such” must be spelled out in some way, and whichever way this is done, it should serve to separate pure general logic from the various special logics. Since special logics apply to certain kinds of entities only, and this is their distinguishing feature, they cannot be oblivious to the differences between objects and their properties. 504 Ch. 5 Logic as the Universal Science II

This in turn suggests that formality-as-particularity should emerge as the key notion in an explanation of the import of “as-such”, and formality-as-normativity, rather than being basic, would then have to be regarded as being dependent upon another concept of formality. However, there may be a relatively simple way past this difficulty. For one might suggest that in the Kantian context “thought-as-such” can be explained with the help of the notion of generality; this is of course suggested by Kant’s official title for formal logic, i.e. “pure general logic”. The suggestion is that “thought-as-such” means all thought; since “all thought” really does mean what it seems to mean, the point behind Kant’s description of general logic as the science which “abstracts from all content of cognition” would be no more (and no less) than that this kind of logic applies to every thought (every judgment, inference, etc.) To this the further characterization can be added that pure general logic applies independently of subject- matter (content, etc.), but the present point is that such further statements do not really add anything to the original description that was formulated in terms of generality. All thought, then, is accountable to the laws of pure general logic; whether we are thinking about mathematics or morals or something else, our thought cannot be at the same time both true and not to be true. This idea sounds straightforward enough. What is more, it can be backed up by arguing that whatever further content “all thought” has surpassing the account given in terms of simple generality, that additional content derives from the fact that “all thought” is being considered in the Kantian context, i.e., against the background of his complex views on the conditions of human cognition. In other words, admitting that pure general logic is formal not only in the normativity/generality sense but also in other senses, it could nevertheless be argued that these other senses are not a part of Kant’s definition of formal logic but substantial consequences that follow, when the definition is combined with his other commitments. This conclusion is also MacFarlane’s. He argues (2000, Ch. 4) that Kant’s general logic is formal in all the three senses given above. Of Ch. 5 Logic as the Universal Science II 505 these, the first – normativity-generality – is the basic or most fundamental one and is what Kant means when he says that pure general logic is formal. That logic in this sense should then turn out to be formal in the other two senses – that logic should turn out to abstract from all differences in its objects and deal with “nothing but the mere [logical] form of thought” (A54/B78) – can be explained by citing his views on concepts, objects and judgments. An argument for the multiple formality of pure general logic, though nowhere given by Kant himself, follows from premises that are uncontroversially Kant’s.17 The argument depends upon a characteristically Kantian assumptions about the connections between sensibility, objects and content:

(1) Objects can be given to us only through sensibility [a basic assumption by Kant, establishing a connection between objects and sensibility]

(2) For concepts to have content, there must be objects falling under them [another basic assumption by Kant, establishing a connection between content and objects]18

(3) For concepts to have content, they must be related to the sensible manifold [from (1) and (2)]

(4) Pure general logic abstracts from the relation of thought to sensibility [consequence of the definition of pure general logic]

(5) Pure general logic represents thought independently of its relation to objects (pure general logic “has no objects”) [from (1)

17 For MacFarlane’s exposition of the argument, see MacFarlane (2000, Ch. 4.4) and (2003, pp. 49-53); the formulation which will be given here is slightly different, but the differences are inessential. 18 More precisely, a concept has content only if the possibility of being given an object or objects is reflected in that concept. 506 Ch. 5 Logic as the Universal Science II

and (4)]19

(6) Pure general logic abstracts from the content of thought [from (2) and (5)]

Abstracting from the object-relatedness of thought and therefore from its content, general logic considers only its form. General logic is thus formal in a sense that goes beyond mere generality. Firstly, it is formal in the sense of non-particularity, as it abstracts from the particular identities of objects and their properties. That is why it is not a special logic, as it abstracts even from the most general conditions of objecthood (these constitute the subject-matter of transcendental logic). Secondly, it is formal in the sense of non-substantiality; since the possibility of being given an object is a necessary condition of having content, according to Kant, pure general logic does not have content. This conclusion and the argument leading to it articulate the characteristically Kantian version of the analyticity-constraint. Since pure general logic is indifferent to the conditions of objecthood and has therefore no content, it cannot make any substantive claims about the world. In other words, Kantian formal logic lacks content in the following two senses:

19 It is useful to be clear about what (5) means. That general logic “does not have objects” or is not object-related means only that it disregards the special conditions on human cognition that stem from the necessarily sensible character of human intuition. It does not mean that formal logic has no bearing on my thought that, say, the Eiffel Tower weighs 7300 tons. Pre- cisely because it is general, pure general logic applies to this particular thought of mine, as it does to any other thought. That general logic abstracts from sensibility and object-relatedness means simply that these facts about human thought are not reflected in its rules and principles; even if, as is impossible, I could think non-sensible thoughts, such laws as the law of non- contradiction or modus ponens would still apply to them exactly as they apply to my actual thoughts. Ch. 5 Logic as the Universal Science II 507

[No-content1] The logical forms recognized in pure general logic are insensitive to the relation that thoughts bear to objects and ignore the distinction between objects and properties. In the predication “S is P”, both S and P are concepts and “is” expresses the relation of containment; P is contained in S, forming part of its intension.

[No-content2] The rules and laws of logic do not state anything about the world; they impose restrictions on thoughts, not on the world.

5.2.3. Analyticity and Apriority

The consequences of the analyticity-constraint for apriority are relatively straightforward. Many commentators have argued that in the case of synthetic apriori propositions Kant fails in the explanatory task that he sets to himself; he starts from the assumption that apriori propositions must be strictly universal and necessary, but ends up explaining them in a manner that renders these properties merely relative. The question whether this criticism is to the point was discussed in section 3.6.3. There it was seen that there are in fact good reasons to think that the notion of necessity that emerges from the conditions of sensibility is absolute rather than relative; for these conditions are best understood as being constitutive of the notion of object simpliciter, rather than any less general notion which would be essentially relative to something else, say, our experience or cognitive make-up. Whether this reply provides a reasonably faithful reconstruction of Kant’s views and whether it suffices to undermine the standard objections are important further questions, but the present point is that in those cases where such conditions are missing, as in the case of pure general logic, there is no reason to think that the rules are anything else but absolute or “strictly necessary” and “strictly universal”, as Kant might have put it. 508 Ch. 5 Logic as the Universal Science II

It seems then that Kant’s account of analytic propositions or analytic judgments is capable of fulfilling the conditions concerning necessity and generality that Russell thinks are indispensable for an acceptable theory of the apriori. Two problems remain, however. The first difficulty has to do with the truth of analytic propositions, the second with the ground of necessity and universality (and truth) that is assumed in Kant’s model of the analytic apriori. The first problem is just the recognition that even though analytic judgments can be called true, they constitute an entirely uninteresting class of truths. For Kant this was not really a problem at all, but simply an admission that the analyticity-constraint was valid for formal logic. The principle of contradiction, which is the “the highest principle of all analytic judgments”, is capable of serving as a positive ground of truth in the case of analytic truths, but this is only because their content is trivial.20 Since analytic judgments are explicitly or implicitly identical, their truth can be recovered by means of this principle, given suitable definitions, but they cannot be used in the extension of knowledge (as Leibniz had apparently thought). Thus, in the case of judgments that are not analytic, the principle of contradiction serves merely as a negative criterion of truth; this function is secured by the logical fact that a contradiction, if found in a judgment, “completely cancels and invali- dates” it (A151/B191). The three conditions on apriority are thus satisfied by Kant’s conception of analytic propositions; they are true and their necessity and universality are strict or absolute and not merely relative. The downside is that this latter feature is secured only by discarding content. And here it must be kept in mind that in the Kantian context “content” does not mean empirical content; rather, it means the content that goes together with the conditions of objecthood, when this is taken in the most general sense. The consequences of the analyticity-constraint are thus quite radical. Ideally, one’s model of the apriori should be explanatory. In the

20 See Kant’s discussion of this principle at the beginning of the Analytic of Principles (A150/B190-A153/B193). Ch. 5 Logic as the Universal Science II 509 case of analytic judgments, Kant pays little attention to this aspect, presumably because there is ostensibly so little to explain. After all, “all bachelors are unmarried” is an evident truth, or can be turned into one, if need be. Its epistemic status – its apriority – is also unproblematic, and only needs to be pointed out. Pointing it out, however, is not yet to explain how it comes about that it has this status. It is here that we meet the second problem. For as soon as one begins to reflect on the possible sources of the analytic apriori that are available to Kant, there arises a difficulty that is exactly analogous to the one besetting the r-model: the source ought to be compatible with the conditions on apriori. No doubt, Kant himself believed that his theory fulfilled this requirement; after all, this theory exhibits the rules and laws of pure general logic as a function of the understanding. A critic, though, might raise a doubt on this point. I shall return to this point below, in section ?? in which the r-model is discussed. The first problem, concerning the relation that formal logic bears to content, will be one of the key issues, when we turn to a consideration of the philosophical underpinnings of Russell’s conception of logic.

5.3. The Bolzanian Account of Logic

5.3.1 Preliminary Remarks

Above in section 5.2.2 I identified two senses in which formal logic has no content, according to Kant. The essence of his argument for this conclusion is simply this. Judgments have content only insofar as they are related to objects. Pure general logic – that is, Kant’s formal logic – abstracts from the fact that our thought is directed towards objects. Therefore, pure general logic has no content. Underlying this argument, there is a quite general assumption to the effect that the elements constituting a thought can be strictly divided into two kinds by some suitable criterion. In Kant’s case, for 510 Ch. 5 Logic as the Universal Science II instance, the criterion has to do with whether the element is constitutive of the relation that judgment bears to objects or not. The division, furthermore, is thought of as explanatory; what it explains and what consequences it has is something that depends upon how the division is understood. Thus there is the following, quite abstract distinction between form and content:

(**) The elements of thought can be divided into those that constitute its content and those that constitute its form. The distinction is explanatory, the explanatory purpose depending upon how the distinction is drawn.

Philosophers who accept (**) often assign logic to form in this abstract sense. This move yields the following no-content thesis:

[no-content3] The elements of thought can be divided into those that constitute its content and those that constitute its form; logic is concerned with form in this sense; therefore, logic has no content.

Since [no-content3] is an abstract characterization of form and content, the other two no-content theses, [no-content1] and [no- content2] are independent of it.21 Philosophers who have accepted [no-content3] have often connected it with [no-content2], but the inference from the former to the latter depends upon substantive assumption about form and content; Kant and logical empiricists can be used to illustrate this point; since form/logic can be established independently of what is given in intuition, it does not assert anything about the world. Yet, there exists at least the abstract possibility that

21 Recall the two other no-content theses. [No-content1] says that logical form is insensitive to the relation that thought bears to objects and ignores, therefore, the distinction between objects and their properties. [No- content2] says that the rules of logic do not state anything about the world; they impose restrictions on thought, not on world. Ch. 5 Logic as the Universal Science II 511 form, understood as in (**), should reside in the world; if that should turn out be the case, it would not follow that logic does not inform us about the world. On the other hand, if one rejects (**), one is thereby committed to rejecting [content2], or the idea that logic is not about the world. At least this follows, when (**) is combined with a natural assumption about the nature of thought. Thoughts are characterized by aboutness; therefore, at least some among the elements of thought must by explained in a way that connects them with this feature. If, now, all elements of thought belong with its content, that presumably implies that they share a common function, and at least a part of this function is to be understood with the help of the notion of aboutness. Even though quite abstract, (**) is nevertheless a substantive principle. What I shall call the Bolzanian account of logic is characterized by its rejection of (**).22 The single most important observation that can be made about the early Russell’s conception of logic is that he accepted the Bolzanian account of logic, which is grounded in the rejection of (**). It is from this standpoint that we can understand the differences between his views and those of Kant’s. The Bolzanian account of logic is important both historically and systematically. In the present context its importance is based on the following facts. Firstly, it has often been formulated in conscious opposition to Kantian philosophy.23 Secondly, Russell’s views on logic constitute a clear instance of it. Thirdly, the model-theoretic conception is a more recent version of the same conception.

22 “Bolzanian”, of course, derives from Bolzano, to whom must go the credit for being the first to work out a detailed version of this conception; this is found in the early sections of his Wissenschaftslehre of 1837. 23 This applies not only to Russell, but to Bolzano as well; see Coffa (1982), (1991, Ch. 2); Proust (1989, sec. 2) for an initial account. For a book- length study of the topic, see Laz (1993); I have not consulted this work, though. 512 Ch. 5 Logic as the Universal Science II

5.3.2 The Basic Assumptions behind the Bolzanian Account of logic.

The basic assumptions that define the Bolzanian account are the following ones. (1) Rejection of the distinction between form and content as an explanatory principle. This is the most important element in the Bolzanian account of logic. As we shall see, an advocate of this account can still draw the distinction, but it serves at best a secondary role. This assumption can be formulated semantically by saying that a semantics which incorporates this non-distinction does not recognize a division of expressions into form-words and content-words. This does not mean that in the relevant semantics there could not different semantic categories, but it does mean that, if there are, they do not reproduce the distinction between form and content. To illustrate, consider Frege’s division of expressions and their semantic correlates into complete (those that stand for objects) and incomplete (those that stand for functions). The sign for identity, for instance, is a logical sign, according to Frege. Yet, it stands for a relation between objects that is just like any “material” relation (say the relation of being taller than). A similar view is also accepted by the early Russell: for example, the sign for material implication, “… …”, stands for a relation between terms and is in that respect like any other relation; it can be distinguished from other relations by indicating what kind of entities it connects, but this is a perfectly general fact that applies to all relations. The non-distinction can also be formulated in ontological idiom – this would be closer to the early Russell’s manner of expressing himself – by saying that on the view under discussion the constituents of (Russellian) propositions (states of affairs) are not divided into those that belong to its content and those that belong to its form. Instead, all constituents belong with content, irrespective of their function. (2) The concept of logical constant is fundamental to the Bolzanian account of Ch. 5 Logic as the Universal Science II 513 logic. In logical tradition the distinction between form and content is often connected with the notion of logical constant. For instance, scholastic logicians included our “logical constants” among syncategorematic terms. These are terms that do not occur as subjects or predicates in propositions and hence do not have signification on their own, but signify only with other (i.e., categorematic) terms. Syn- categorematic terms in turn belong with the forms of propositions, as opposed to their matter (or content).24, 25 Thus, scholastic logicians said, for example, that formal consequence is consequence in virtue of form, or consequence that depends for its validity or truth only upon the presence of the relevant syncategorematic terms.26 Even if the distinction between form and content is given up, an advocate of the Bolzanian account can still say that there are “logical constants”. Indeed, logical constants may be of special importance. For example, Bolzano himself was particularly interested in logical consequence (roughly, consequence in virtue of logical constants). And for Russell the division of constants, or constituents of propositions other than variables, into logical and non-logical was essential. Logicism for him was the thesis that the constants occurring in mathematical propositions are all of them logical in character. Clearly,

24 Of course, the scholastic use of “term” is quite different from the early Russell’s; scholastic terms belong to language, natural or mental, whereas for Russell, “term” is equivalent with the notion of any entity whatsoever. 25 In Jean Buridan we find the following formulation: “I say that in a sentence in which we speak of matter and form we understand the matter of the consequence or sentence to be the purely categorematic terms, namely the subject and the predicate. The syncategorematic terms added to it, by which the subject and predicate are connected or denied or distributed or taken to supposit in a certain way are not included. And we say the entire remaining part of the sentence pertains to the form.” (Jean Buridan, Treatise on Conse- quences (Buridan (1985, Ch. 1.7.2/p. 194) 26 The connection is very clearly explained by Jean Buridan. According to Buridan, formal consequence is one that is valid for uniform substitutions of categorematic terms (“A consequence which is acceptable in any terms is called formal, keeping the form the same” (Buridan 1985, Ch. 1.4.2/p. 184). 514 Ch. 5 Logic as the Universal Science II such a view presupposes that there is available some principled criterion for the identification of constants that are said to be logical. This notion is therefore fundamental to Russell’s thinking about logic. Although the distinction between form and content can be retained, the way it is drawn brings out clearly the importance, precisely, of logical constants. This point is illustrated by Bolzano’s theory of consequence. Bolzano pointed out that we can study a range of important logical properties of propositions by considering “in a given proposition not merely whether it is itself true or false, but also what relation to truth follows for all the propositions that develop out of it when we assume certain of the ideas present in it to be variable and permit ourselves to exchange them for whatever other ideas” (Bolzano 1837, §147). With the help of this device of variation – keeping some parts of a proposition fixed or constant while varying the others – he was able to define several important concepts. First, he defined universal or complete validity, a notion that is always relative to some of the propositions constituents, by saying that a proposition is universally valid (universally invalid) with respect to its constituents a1...an only if every proposition that results from admissible substitutions of other “ideas” for a1...an is true (false) (ibid.) Bolzano then said that a proposition is analytically true if there is at least one constituent in it with respect to which the proposition is universally valid (id., §148; analogous connection holds between analytical falsity and universal invalidity). As a special case of analytic propositions he mentions that one can imagine all the logical constituents of a proposition to be fixed; in that case one would have before one’s eyes the notion of a logically analytic proposition, proposition such that every admissible substitution for its non-logical constants yields a true (or false) proposition. Switching from truth to consequence, one can define a general notion of (analytic) consequence as follows:

“I say that propositions M, N, O ... would be derivable from propositions A, B, C, D ... with respect to the variables i, j,... if every set of ideas which makes A, B, C, D ... all true when substituted for i, j, ... also makes M, N, O, ... all true.” (id., §155) Ch. 5 Logic as the Universal Science II 515

In other words, and simplifying somewhat, a proposition is “derivable from” (or “can be inferred” or “concluded from” (ibid.)) a collection of premises with respect to a number of its constituents, if variation of constituents never leads from true premises to a false conclusion.27 In Bolzano’s definition of consequence it is thus assumed that some parts of propositions have been declared constant and other variable or subject to variation. With right choice of constants, this procedure yields a very reasonable notion of logical consequence, as several commentators have pointed out. What, then, of form and content? Bolzano was not very happy with his predecessors’ attempts to delineate the sphere of logic with recourse to the distinction.28 And yet he is willing to admit that there is a perfectly clear sense in which logic can be called a formal science, a sense that can be brought out by connecting it with the idea of variation.29 This means that for Bolzano the distinction between form and content is not self-contained or explanatory, but depends upon a prior provision of specifically logical constants. MacFarlane (2000, Ch. 2.2) uses the term “schematic” to refer to this notion of formality. He calls it a “decoy notion of formality”, meaning that it is insufficient on its own for demarcating logic, and must therefore be completed, among other things, with a criterion of logicality, or logical constants. Bolzano was well aware of this feature of his account of consequence.30 In §148 of Wissenschaftslehre, having distinguished the

27 For the details of Bolzano’s concept of consequence, see, e.g., Berg (1962), Sebestik (1992), Siebel (2002). 28 It is not an insignificant fact that most of these predecessors were Kantian philosophers; for Bolzano’s discussion, see (1837, §7). 29 In this sense, logic deals with kinds of propositions rather than particular propositions, where, for instance, “Some As are Bs” forms a kind, with A and B indicating the content (or matter) of propositions and the rest their form. It does not follow, however, that logic would be purely formal science (as the Kantians were inclined to say); logic still has its own matter and its own characteristic truths; see §12 of Bolzano (1837). 30 This is also pointed out by MacFarlane (2000, p. 41). 516 Ch. 5 Logic as the Universal Science II set of logically analytic propositions from other analytic propositions, he adds the cautious comment that “this distinction has its ambiguity, because the domain of concepts belonging to logic is not so sharply demarcated that no dispute could ever arise over it”. Again, in §186 he writes that since there are indefinitely many properties of propositions which can be studied by means of the method of variation, this method is bound to be incomplete as a characterisation of logical form; the ideas relevant to logic, though formal in the sense explained, must be singled out in some other way.31 (3) Existence of a non-arbitrary criterion for logical constanthood. As the above discussion indicates, it is a natural assumption that the entire Bolzanian account of logic depends for its viability upon a non- arbitrary notion of “logical constant” or criterion for logicality. At any rate, this appears to be how the advocates of the account have seen the matter. Typically, they have some philosophical project in mind in which a criterion for logicality is needed. Had Russell simply stipulated that such and such notions were logical, the logicist project could not have had the philosophical significance that he attributed to it. An- other example would be the notion of logical consequence itself. In- sofar as logic is thought of as the discipline that is concerned with the inferential correctness of reasoning, a proposed definition or explication of consequence remains accountable to at least some of our intuitions concerning consequence. The alternative would be to regard the definition as introducing a new notion by stipulation. Given a suitable context, that could be a perfectly reasonable attitude towards one’s concept of consequence. Such contexts, however, were alien to philosophers like Bolzano or Russell.32 Looking for a suitable criterion of logicality, we can turn to the three notions of formality introduced in section 5.2.2 for an explica-

31 Tarski’s well-known remarks on logical consequence in his (1936) are exactly analogous to Bolzano’s not only in their content but their context as well. 32 For a different attitude, cf. Tarski’s letter to Morton G. White; Tarski (1987). Ch. 5 Logic as the Universal Science II 517 tion, not of formality, but of logicality or logical constanthood. Evi- dently, the third notion, non-substantiality, can be set aside. For it implies the view that logic abstracts from the relation of thought to world. This view, however, is precisely what the Bolzanian conception was designed to undermine in the first place. Thus we are left with the first two notions, normativity and non-particularity as candidates for explications of logicality or logical constanthood. A summary of the Bolzanian account of logic is worthwhile at this point. The following three elements were distinguished in it. Firstly, it rejects the distinction between form and content, insofar as this implies (i) the idea that logic has no content, but is purely formal, and (ii) the explanatory primacy of the notion of form, or the idea that logic is a distinctively formal discipline. Secondly, instead of form, the Bolza- nian conception focuses on the notion of logical constant; this is shown very clearly by the schematic notion of form.33 This promises explications of such notions as logical consequence and logical truth, and does so by relying on antecedent provision of expressions or entities that qualify as logical constants. Thirdly, the notion of logical constant must receive a non-stipulative characterisation; for this purpose, one may turn to such ideas as normativity and non-particularity. No doubt, other candidates for this role are available, but these are the relevant ones in the present context.

5.3.3 Russell’s Version of the Bolzanian Account of Logic

5.3.3.1 Russell on Form and Content

It was mentioned above that Russell’s conception of logic is an instance of the Bolzanian account. I shall begin examining Russell’s version by considering the notion of semantic content that was intro-

33 This use of “schematic” must be kept separate from the notion of schematic conception of logic that was used in section 4.2.1, where the technical core of the model-theoretic conception of logic was described. 518 Ch. 5 Logic as the Universal Science II duced in section 5.2.2. The view that logic is formal is often paired with the further view that it says nothing about the world. In Kant’s case that was seen to follow from the claim that logic is formal in the sense of being normative for all thought; since there is a sense of thought that is intelligible apart from its relation to objects, norms that are valid for all thought must abstract from this relation; therefore, these norms do not impose constraints on how the world is but only on our thought. “Semantic content” is thus related to [non- content2]. It is important not to confuse the notion of semantic content with that of particular content. That logic abstracts from particular content is an idea that Russell often resorts to when he needs a short and intelligible way of describing the subject-matter of logic. Although famous for his changes of mind, this idea is stable element in his thinking about logic. It is found, for example, in “The Philosophical Impor- tance of Mathematical Logic”, a paper written in 1911:

Now, in a deduction it almost34 always happens that the validity of the deduction does not depend on the subject spoken about, but only on the form of what is said about it. Take for example the classical argument: All men are mortal, Socrates is a man, therefore Socrates is mortal. Here it is evident that what is said remains true if Plato or Aristotle or anybody else is substituted for Socrates. We can, then, say: If all men are mortal, and if x is a man, then x is mortal. This is a first generalization of the proposition from which we set out. But it is easy to go farther. In the deduction which has been stated, nothing depends on the fact that it is men and mortals which occupy our attention. If all the members of any class ơ are members of a class Ƣ, and if x is a member of the class ơ, then x is a member of the class Ƣ. In this statement we have the pure logical form which underlies all the deductions of the same form as that which proves that Socrates is mortal. (Russell 1911a, p. 35)

On the view outlined here, logic is not concerned with particular objects (like Socrates or Plato or Aristotle) nor with particular properties and

34 I do not know why Russell says “almost always” rather than simply “always”. Ch. 5 Logic as the Universal Science II 519 relations (like humanity and mortality), but strives for greater, indeed, maximal generality. A “principle of deduction” like the Barbara syllogism applies with equal force to any object and property that one ca- res to mention. Hence, insofar as one’s concern is with pure, rather than applied logic, no particular object or property should feature in the formulation of its laws and principles. Even if logic abstracts in this way from particular content, it does not follow from this alone that logic should be entirely without content. Indeed, the process of abstraction or generalization or purifica- tion that Russell describes in the quotation yields results like “If a thing has a certain property [belongs to a certain class], and whatever has this property [belongs to this class], has a certain other property [belongs to a certain other class], then the thing in question also has that other property [belongs to that other class]”. A proposition like this does not seem to be at all without “content” in some intuitive sense of that word. As we saw in section 5.3.1, in order for the no content-thesis to follow, one needs to assume, among other things, that the elements featuring in a thought (proposition, etc.) can be divided into two mutually exclusive classes: (i) those that belong to its content, and (ii) those that belong to its form. The later Russell accepts this assumption, as is shown by the following passage from the manuscript “What is Logic?”, composed in 1912:

In a complex, there must be something, which we may call the form, which is not a constituent, but the way the constituents are put together. If we made this a constituent, it would have to be somehow related to the other constituents, and the way in which it was related would really be the form; hence an endless regress. Thus the form is not a constituent. (Russell 1912b, p. 55; italics in the original)35

35 See also Russell (1913, pp. 97-98) and (1914a, p. 52). It is, of course, somewhat dangerous to speak of “the later Russell” who accepts this or that view. In this case, however, this is justified. In general, one reason why the notion of logical form began to play an increasingly important role for Rus- sell was that he came to think there is a connection between that notion and 520 Ch. 5 Logic as the Universal Science II

The later Russell thus accepts the following distinction among the elements of propositions (or elements of complexes, i.e. facts):

CONTENT: non-logical constants FORM: logical constants + variables36

Since the propositions of logic do not contain any non-logical constants, it is an immediate consequence of this account that they have no content in this sense. As we have seen, this does not yet imply that they should have no content at all. Whether it does depends upon the problem of unity. As the quotation from “What is logic?” shows, he thought of logical form, among other things, as the way the constituents of a proposition (complex) are put together. It is therefore quite natural to think that these “ways” have something to do with the unity of a complex. Re- flecting on this problem, he came to think that that which is responsible for unity cannot itself be a constituent of a complex; hence the idea that the elements of thought (complex, proposition) must be separated into those that belong to its content and those that belong to its form. 36 Here I will ignore the problematic nature of variables. It is not clear whether the later Russell regards them as mere symbols or as both symbols and something that variables in the linguistic sense stand for. It is, perhaps, difficult to make clear sense of the latter idea, but it would seem that, for Russell, they must be more than mere symbols. This is indicated by the fact that the forms of what he calls atomic complexes are symbolized by mere strings of variables: for example, the proposition that Socrates is mortal yields the form “Fx”, when non-logical constants have been eliminated from it. If, now, variables are no more than linguistic devices, nothing is left of the form itself. This conclusion is simply incompatible with the bulk of what Russell says about logical forms. Above all, there is the idea, developed in some detail in the Theory of Knowledge-manuscript, that logical forms are objects of acquaintance. Evidently, at this stage, Russell does not mean that our knowledge of logic is purely linguistic (see Part I, Chapter IX of Russell (1913)). Whichever way they are dealt with, there must be something in the form that matches the use of variables: for instance, the form that is expressed by “xRx” must be distinguished from the form “xRy”; for Russell’s ideas on this point, see Part II, Chapter I of the Theory of Knowledge - manuscript as well as chapter XVIII of Russell (1919). Ch. 5 Logic as the Universal Science II 521 how forms are understood. Insofar as a philosopher accepts the analyticity-constraint, he is committed to the view that there is a strict division of the elements of a thought into those that belong to its form and those that belong to its content. Given suitable further assumptions about the source of “form”, it follows that logic has no (semantic) content. The logical empiricists, who were followers of Kant on this point, argued that the form of “thought” (the language of science) is attributable to purely conventional rules (at least this was the view that Carnap held in the 1930s), where conventionality is meant to cancel out the impact of the world on thought; the choice of form is thus up to us, and is not constrained by worldly facts; this is Carnap’s principle of tolerance. The later Russell is a good deal less transparent about the matter. He does accept that there is a distinction between form and content, when he argues that the way in which the constituents of a proposition or complex are put together is not itself a further constituent. Nevertheless, it is not clear what consequences he drew from this. There are passages which might be taken to indicate that he accepted the no semantic content-thesis. For instance, in Our Knowledge of the External World he argues that, since the formal proposition which underlies “If Socrates is a man, and all men are mortal, then Socrates is mortal” “does not mention any particular thing, or even any particular quality or relation, it is wholly independent of the accidental facts of the existent world, and can be known, theoretically, without any experience of particular things or their qualities and relations” (1914a, p. 67). Yet, this establishes at most that logic is independent of what is contingently the case in the existent world; it does not show that logic is not factual, or that no experience is needed to establish logic. And if we recall his famous statement that “logic is concerned with the real world just as truly as zoology, though with its more abstract and general features” (1919, p. 169), we might conclude from this that for Russell logic is substantial in some fairly robust sense. No doubt, the correct conclusion is that his views on this matter were not settled. This, however, is a problem for the later Russell. As regards the 522 Ch. 5 Logic as the Universal Science II

Russell of the Principles, this kind of contrast between form and content simply does not exist: all elements of a proposition belong to its content. For the early Russell, then, the picture is simply this:

CONTENT: non-logical constants + logical constants + variables

As we shall see below, even the early Russell accepts that the propositions of logic abstract from particular content. They nevertheless possess content, namely the content that belongs with logical constants. Logic is therefore not a “purely formal science”, as a Kantian might say, but has its own characteristic material or concepts. The degree of robust- ness of logic is therefore exactly the same as that of any other science. This conclusion, one might argue, is contradicted by Russell’s universalism about logic and his denial of metaperspective. Here the claim is that even if logic has its own characteristic concepts like implication, negation, identity, quantifiers, and so on, the laws of logic are nevertheless not about these concepts in the way that, say, chemis- try or zoology is partly about its own characteristic concepts. To this the following reply should be made. True, the universalist does regard the statements of logic as generalizations made by dint of the characteristically logical items, rather than statements about these items; to that extent, logic cannot be said to have a subject-matter in the same way as other sciences do. This, however, does not show that logical concepts are not “material”. Firstly, it is clear that they do make a difference; if in the inference “All men are mortal, Socrates is a man Socrates is mortal” we replace “all” by “some”, the inference ceases to be logically correct. Hence “all” does have semantic import. No doubt, this is less than conclusive, as one can presumably reconcile the view that logical items have semantic import with the view that they belong to form and not content. Secondly, and more importantly, it is very difficult to understand the universalist construal of logical laws as generalizations about objects, properties and relations in a way that makes these generalizations “purely formal” in the sense Ch. 5 Logic as the Universal Science II 523 of not having content.37 On the universalist conception of logic, the status of logic is not different from that of other, “special” sciences. Russell does not state this conclusion in as many words as Bolzano or Frege, who shared the view with him.38 Nevertheless, the essentials of Russell’s logical framework are so similar to Bolzano’s or Frege’s that the conclusion easily follows. Above all, there is Russell’s proto-semantics, or the view that the meaning of a linguistic expression is the entity which it stands for. This simple “Fido”-Fido -model for semantics assigns a representative function to all semantically significant parts of a sentence – and, as a consequence of this, to the sentence itself, which stands for or “expresses” a proposition, which is semantically speaking just a complex of “meanings”. The semantics of logical words, insofar as these are recognized as a separate class of expressions, is therefore explained in exactly the same way that one explains the semantics of any other kind of expression.39 Or, again, we can give a metaphysical formulation of this point, saying that, for Russell, logical constants are constituents of a proposition in exactly the same way as its non- logical constants. For instance, he writes at the very beginning of the Principles that the propositions of pure mathematics do not contain “any constants except logical constants” (§1); this statement, I submit, is to be understood literally. On the Bolzanian account of logic, we have seen, one can retain a distinction between form and content, although not for explanatory purposes. This applies to the early Russell, too. In Part I of the Princi- ples there is plenty of talk about form and formality. Closer inspection shows, however, that his understanding of the distinction accords with the Bolzanian account. The adjective “formal” is repeatedly used by Russell in the first part of the Principles, where the indefinables of mathematics are exam-

37 For more on this, see section 5.7.2 below. 38 For Bolzano’s views, see section 5.4.2; Frege discusses the view at length in his (1906, pp. 338-339). 39 Cf. here Proust’s comments on Bolzano: Proust (1989, pp. 54-6). 524 Ch. 5 Logic as the Universal Science II ined. To begin with, the logic of mathematics is referred to as “For- mal Logic”, though Russell’s standard term for it is “Symbolic” rather than “Formal” (see Chapter II of the Principles). This, of course, may be no more than a gesture in the direction of tradition; the science which tradition knows as “formal” (or “symbolic”) logic can be shown to constitute the foundation of mathematics, once its scope has been correctly understood. It seems, however, that Russell’s use of “formal” is not purely nominal in this way. Firstly, Russell criticizes Kant for thinking that reasoning in mathematics is “not strictly formal” but uses intuition (§4). As against Kant’s view, Russell argues that, thanks to the work begun by Peano, all mathematics can be “strictly and formally deduced” (ibid.). Deduc- tion is thus labelled “formal” by Russell, and this characterization is meant to highlight the difference between his own views and Kant’s. Secondly, if deduction is formal, one might surmise that the principles underlying deduction are formal, too. And this is, indeed, explicitly stated by Russell, when he calls the laws of addition, multipli- cation, negation and tautology “formal” (§25). There is, though, one principle of inference that is incapable of a “formal symbolic statement”; this is Russell’s version of the rule of detachment (see Princi- ples, §§18 and 45). The existence of this non-formal principle illustrates, according to Russell, the “essential limitations of formalism” (§18), but it does not show that deduction per se would be non- formal.40 The formal character of deduction is further indicated by the fact that all deduction is based, in the last instance, on suitable formal implications;41 since formal implications are to be distinguished from material implications, one might conclude that the former hold in virtue of their form, while the latter are implications that hold in virtue of content (and eventually, truth-values). Whatever else may be characteristic of formal implications, they at least hold for every term. And this, Russell writes in §45, is the essence of what may be called

40 For further discussion, see section 5.9. 41 Cf. §45 of the Principles, where the dependence is stated explicitly. For further discussion, see, again, section 5.9. Ch. 5 Logic as the Universal Science II 525 formal truth. Not only implications and truths are formal. Properties, too, can be formal. And we find Russell arguing that pure mathematics is throughout concerned with such formal properties. The relation of temporal priority, for instance, falls outside the sphere of pure mathematics. What matters to pure mathematics are the formal properties of this relation and any similar relation, properties summed up in the notion of (ordinal) continuity. There are more than one way of characterizing what a formal property is, but one way of doing so is to say that they are properties which give rise to formally identical deductions (cf. §§8 and 12 of the Principles); Russell’s terminology may be non-standard, but the underlying idea is familiar: since, for example, the formal properties of complex numbers are the same as those of points in a Euclidean plane, same deductions apply to both of them (ibid.) It might be thought that all this talk of formal deductions, formal implications, formal truths and formal properties means that Russell did, after all, think that logic has no content in the sense of [no- content3]. This, however, would be a mistake. The formality that he has in mind is not of the sort that deprives the propositions of logic expressing these formal notions of their semantic content. At most one can say – and Russell himself says as much – that logical propositions are without particular content; as we have seen this is something quite different from saying that they have no content at all, or that they have no semantic content. To make a case for this claim, two things will have to be established. Firstly, it must be shown that Russell’s talk of “form” accords with the Bolzanian account of form. Secondly, on the Bolzanian account, “form” is in the fundamental sense a purely schematic notion. Therefore, it must be seconded with an independent criterion for logicality; this criterion, furthermore, must be one that undermines the no content-thesis, rather than supports it. 526 Ch. 5 Logic as the Universal Science II

5.3.3.2 Russell’s Version of the Schematic Account of Logical Form

In order to discuss the schematic notion of logical form in the context of Russell’s philosophy, we need a more detailed account of what is involved in that notion. It is simplest to formulate a linguistic variant of that notion, although historically such an approach was a late- comer.42 “Logical form” is a notion that is introduced for the purpose of defining such concepts as logical truth and logical consequence or logically valid argument. First, we need a couple of assumptions about the language for which the definitions are formulated:

(1) logical truth and logical consequence are defined for a fully interpreted language, L;43 (2) the lexicon of L, lex(L), is divided into the subset f of fixed terms, and v of variable terms; (3) every member of v belongs to one and only one grammatical- cum-semantic category, which is a class of intersubstitutable terms;

Next we explain the logical properties of sentences of L. For this purpose we need a class f of constant terms, and, for every sentence, S, of L, the notion of “associated class of sentences”, S(f). A member of S(f), S´, is a sentence of L such that S´ can be derived from S – or S can be transformed into S´ – by a series of permissible substitutions, which leave the fixed terms occurring in S intact; to be permissible, a substitution must be effected within a semantic category and it must

42 The following account draws on Etchemendy (1983, sec. IV). What I have called the schematic account Etchemendy calls the substitutional/interpretational account of logical properties; see also his discussion of interpretational and representational semantics in Etchemendy (1990). 43 It is a part of this full interpretation that the domain for the language has been fixed; for now, though, this feature of the account is not important. Ch. 5 Logic as the Universal Science II 527 be uniform. To illustrate, suppose that f is the set {“not”, “or”} and S is the sentence “snow is white or snow is not white”. S can now be transformed into other sentences of L: these, as we might say, agree syntactically with S but differ from it with respect to the members of v. For instance, S can be transformed, lexicon permitting, into such sentences as “snow is yellow or snow is not yellow”, “grass is pink or grass is not pink”, etc. We can now say that the original sentence, S, is logically true with respect to the constants constituting f, if and only if every member of its associated class S(f) it true. Thus, “snow is white or snow is not white” is logically true with respect to f = {“or”, “not”}, because every sentence that can be derived from it, or every sentence into which it can be transformed, by admissible substitutions is true. On the other hand, if f did not contain the word “or”, we could have transformed that sentence into a falsehood. For example, we could have substituted “and” for “or”, transforming the original sentence into a falsehood. This shows that S is logically true with respect to {“not”, “or”} but not with respect to {“or”}. Logical truth is thus characterized by the following two features.

(LT1) Logical truth is a relative notion; a sentence qualifies as logically true (or not) relative to a selection of constant terms.

(LT2) Logical truth is defined with the help of the notion of ordinary, material truth.

This account can be extended to logical consequence or logical validity; analogously to (LT2), this is defined with the help of the notion of ordinary or material consequence, which is a matter of simple truth- preservation. We say that an argument is truth-preserving if at least one of its premises is false, or when its conclusion is true,44 and we say that an argument is logically valid, or that its conclusion follows logically from the premise or premises, if every argument belonging to its

44 Cf. here Etchemendy (1983, pp. 326-7). 528 Ch. 5 Logic as the Universal Science II associated class is truth-preserving. The schematic accounts of logical form, logical truth and logical consequence come in different varieties, depending on how one chooses think about some of the details. As for Russell, the most striking thing about his version is that he rejects the linguistic presupposition, which was used in the above account. That presupposition, however, is by no means inherent in the schematic account, and does not therefore suffice to undermine its attribution to Russell.45 The adjustments that this shift in perspective forces upon the linguistic variant are straightforward and need not be scrutinized here in detail; they result from a simple translation from talk of sentences, expressions, syntactic categories, etc. into an ontological idiom of propositions, propositional constituents, and so on.46

45 Giving up on the linguistic presupposition, Russell is merely agreeing with Bolzano, for whom logical theory was concerned with “sentences in themselves” (Sätze an sich). 46 There is, in fact, a very good reason for preferring an ontological formulation over the simple linguistic variant. As Etchemendy (1990, Ch. 2) points out, a simple linguistic variant of the schematic account – what he calls the substitutional test for logical truth (id., pp. 27-33) – has a serious drawback. Suppose our language had only an extremely meagre stock of lexical items in it. It could happen, for instance, that “Abe Lincoln was president” passed the test for logical truth outlined above, because our language did not have any proper names for non-presidents. In that case every member of the associated class of the original sentence (say “George Wash- ington was President”, “Grover Cleveland was President” and “William Taft was President”) would be true, but we would not want to say that the original sentence was logically true. Evidently, an account of logical truth and validity should not be made dependent upon such accidental quirks of the language as whether or not it has proper names for such and such persons. This problem can be side-stepped by denying that logical truth and validity are properties of linguistic items in the first place, thus shifting one’s focus from sentences to suitable non-linguistic items. Whether or not our language has names for non-presidents, there are such entities as Josef Ratzinger’s being a President, and the mere existence of such entities shows that the original claim about Abraham Lincoln is not a logical truth. There is, of course, a more up-to-date way of avoiding the problem. This was adopted by Tarski Ch. 5 Logic as the Universal Science II 529

In Russell’s case the schematic account remains largely implicit, but it is nevertheless there. He comes closest to an explicit formulation in Russell (1905a), paper which he read to the Oxford Philoso- phical Society in October 1905, but did not publish. In the paper Russell does not use the language of “variation” as Bolzano had done, but uses the notion of propositional function. Another difference is that the paper is not about the schematic account itself but about finding a serviceable notion of necessity. These differences do not undermine the basic point, however. Trying to identify a reasonable notion of necessity, Russell turns to propositional functions, defining:

$ The propositional function is necessary if and only if it is true for all values of x.47 $ The propositional function is possible if and only if it is true for some values of x. $ The propositional function is impossible if and only if it is not true for any value of x.

One can extend this analysis to propositions by stipulating that a proposition is necessary (possible, impossible), if it is an instance of a propositional function that is necessary (possible, impossible) in the sense just explained. Here it must be taken into account, however, in the 1930s. He introduced the notion of satisfaction, which is a relation between certain linguistic items, namely sentential functions, and non-linguistic items (objects); instead of considering the various sentences or propositions that are associated with “Abe Lincoln was President”, he considered the relation which holds between “x was President” and an arbitrary object if and only if that object satisfies the function (if and only if the object was President). 47 To say that a propositional function is true for all values of the variable is to say that no matter what constant is substituted for the variable, the result is always a true proposition. Russell’s own preferred formulation is to say that the function is “always true” (“sometimes true” and “never true” for the other two cases). 530 Ch. 5 Logic as the Universal Science II that a proposition can always be regarded as a value of several propositional functions. For instance, the proposition /Plato is taller than Socrates/ can be regarded as resulting from , , , etc. Under- stood in this way, necessity, possibility and impossibility turn out to be relative notions. For instance, /if Socrates is human, Socrates is mortal/ is necessary if it is regarded as a value of , but contingent if it is regarded as a value of . Another way of putting the point would be to say that Russell’s account relativizes modal attributions to propositional constituents: for example, the above proposition could be said to be necessary with respect to Socrates. This brings Russell’s account close to the Bolzanian one. It could be said, indeed, that propositional functions are just the technical counterpart of the notion of variation: talk of “function” and “argument” captures the idea that something in a proposition is held constant and something subject to variation. Moreover, when necessity is understood in the way Russell understands it here, there is but a small step from it to the logical properties envisaged by Bolzano. It comes as no surprise, then, when Russell suggests that the propositions of logic could be singled out by the feature that they are “necessary with respect to all their constituents except such as are what I call logical constants” (1905a, p. 519). For instance, we can say, using this terminology, that “Socrates is identical with Socrates”’ is necessary with respect to Socrates; no matter what is substituted for Socrates, the result is always a true proposition, a value of the propositional function “x is identical with x”.48 In “Necessity and Possibility” Russell uses the terminology of propositional functions rather than variation. As we saw in section

48 Using the terminology introduced above, we can say that the propositions which are values of “x is identical with x” constitute the class associated with the proposition “Socrates is identical with Socrates”, when “is identical with” is held constant; “Socrates is identical with Socrates” is thus logically true with respect to {“=”}. Ch. 5 Logic as the Universal Science II 531

4.4.9.3, however, the connection between the two was envisaged explicitly in the Principles. Recall that there Russell conceived of propositional functions as suitable for the technical development of Symbolic Logic or a calculus for logic. At the same time he entertained the hope that philosophical analysis could explain them with the help of the notions of proposition, propositional constituent and variation. And the first step towards such an analysis was to note that “[i]n any proposition, however, complicated, which contains no real variables, we may imagine one of the terms, not a verb or adjective, to be replaced by other terms: instead of ‘Socrates is a man’ we may put ‘Plato is a man,’, ‘the number 2 is a man,’ and so on” (§22). This act of imagination yields a number of propositions associated with the proposition chosen as the starting point. In typical cases, Russell suggests, some members of the associated class are true and others false: in the language of propositional functions this amounts to saying that propositional functions are typically contingent. There are of course, exceptions. In particular, there are the cases where all members of the associated class of propositions are true. It is such cases that Russell singles out for special attention, calling them “formal implications”. Evidently, we have here all the elements of the schematic account of form. The claim that Russell’s use of “form” accords with the schematic account can be further supported by considering the role that formal implications play in Russell’s account of logical inference. He argues – somewhat surprisingly perhaps, but perfectly in accord with the Bolzanian account of logic – that ordinary truth-preservation (cf. above) is the key notion in explaining what valid inference is. I shall return to this issue below (cf. section 5.8.2) but the basic idea is worth mentioning here. It is simply this:

an inference exhibiting a particular form is valid if and only if every argument of that form is truth-preserving.

Here every argument is explained by dint of the notion of formal impli- 532 Ch. 5 Logic as the Universal Science II cation, exhibiting a particular form is explained with the help of he Bol- zanian account of form, highlighting the role of the notion of constant, and truth-preserving is captured by the notion of material implication. Russell explains this all very clearly in §37 of the Principles, though of course without explicit reference to the Bolzanian account. I offer no quotations here, as I shall discuss the issue more fully below. Whatever else may be said about Russell’s account of inference, it is at least clear that it accords with the Bolzanian account. These observations suffice to show that Russell’s use of “form” accords with the schematic account of logical form and logical properties that is the hallmark of the Bolzanian account of logic. We have already noted that the account, being purely schematic, needs to be accompanied by a criterion for logical constanthood (assuming that our interest lies in logical truth and logical validity). In the absence of a reasonable such criterion, the account is vacuous. We can always declare that a particular expression or set of expressions is to be treated as constant. Given a non-standard choice, we might end up saying oddities like “Abe was a US President, therefore Abe was male” is a logically valid argument with respect to {“US President”, “male”}: no matter what is substituted for “Abe”, the result is always an argument that is truth-preserving. But, of course, neither “US President” nor “male” is a serious candidate for being a genuinely logical constant. The advocates of the schematic account have not failed to register this feature of their theory. I quoted above Bolzano’s statement to the effect that one could distinguish among analytic propositions those that are logically analytic, i.e., propositions whose invariant parts are concepts belonging to logic. This is the skeleton notion of logical constant that emerges from the schematic account of logical properties: divide the constituents of a proposition into those that are constant and those that are subject to variation; the first constitute the form of the proposition, and if they all belong to logic, they delineate the logical form of the proposition. Bolzano then goes on to remark that a logically analytic proposition is one that can be judged Ch. 5 Logic as the Universal Science II 533 as analytic on the basis of logical knowledge alone; such characterizations are bound to remain vague, he adds, for the domain of concepts belonging to logic is not so sharply delineated that no controversy could arise over it. Russell’s understanding of the role of logical constants is similar. Although he often uses “formality” the capture the idea of logicality, the real explanatory burden is carried by the notion of logical constant. Most important of all, the propositions of pure mathematics – and, hence, of logic – are propositions containing nothing but variables and logical constants, as Russell explains in §§1 and 8 of the Principles. They are formal in the appropriate sense, but their formality is something that is conferred upon them by the presence, in them, of constants belonging to logic. The question thus arises: How does the early Russell propose to distinguish constants that are logical from constants that are not logical? In other words: What is his criterion of demarcation for logic? In considering this question, we may refer to the notions of formality introduced in section 5.2.2. This time, though, they are not to be understood as providing criteria for formality; instead, they should be understood as explicating the more general notion of logicality. Given Russell’s other assumptions about what logic is and what it is capable of accomplishing, we may safely assume that there are two serious candidates for such explications, not necessarily exclusive of each other.

$ the normative notion of logicality $ logicality as non-particularity

To these we must add the implications of Russell’s universalist conception of logic: what, exactly does it imply, as regards the question of demarcation? In particular, how does universalism about logic relate to the two notion of logicality? 534 Ch. 5 Logic as the Universal Science II

5.4 The Universalist Conception of Logic and Logical Constants

Some scholars have argued that universalism about logic has fairly straightforward implications for the issue of logicality. They have argued, firstly, that a great deal of the content of the universalist conception comes from the view that logic has a special, foundational role to in thought. Secondly, they have argued that this role has far- reaching consequences, when it comes to explicating the nature of logic, at least if this is taken to involve an informative explanation of logical constants; this, of course, is the “logocentric predicament” to which Harry M. Sheffer referred to in his review of Principia: in order to give an account of logic, one has to presuppose and use logic.49 In the Principles there are two passages which bear directly on the definability of logical constants. The first is from §10:

The logical constants themselves are to be defined only by enumeration, for they are so fundamental that all the properties by which the class of them might be defined presuppose some terms of the class. But practically, the method of discovering the logical constants is the analysis of symbolic logic [...].

On the face of it, this passage expresses a view about logical constants that is exactly what the interpretation under consideration argues a universalist about logic should hold: since logical constants are fundamental to all thought, they occur whenever thought occurs, and this includes the provision of definitions. Therefore, they are terms which are used, among other things, in framing definitions of anything, rather than terms that are themselves defined. Yet, a closer look suggests that the point of the passage and the argument leading to it may well be rather less transparent. Russell’s point would be clear enough if the passage were intended to establish that there cannot be informative definitions of particular

49 Cf. section 4.5.2.2.3. Ch. 5 Logic as the Universal Science II 535 logical constants, the reason for this being that such definitions inevitably presuppose the term that is to be defined. This is a familiar point about logical constants, and seems to apply no matter how one formulates one’s definitions of logical constants. Consider for example a model-theoretic definition of a logical constant, which is a statement of the truth-conditions of sentences featuring that constant; for example:

(*) M »= P Q if and only if M »= P and M »= Q.

That is, a sentence of the form P Q is true in a model M if and only if P is true in M and Q is true in M. (*) thus explains the object- language conjunction connective, “ ”, by dint of a clause that is itself conjunctive. It is a reasonable assumption that conjunction is semantically something so fundamental that there is no explanation of it in more basic terms: grasping (*) presupposes that one already grasps the semantic content of conjunction, i.e., the concept which is expressed in the object-language by “ ” and in the metalanguage, in this case English, by “and”. This kind of circularity, however, is not harmful, for the function of (*) is merely to explain the semantics of “ ” to someone with a sufficient grasp of the metalanguage. It could be argued that this escape route is not open to the universalist, for he is supposed not to distinguish between object- and meta-languages; and even if he does, the distinction does nothing to undermine the view that insofar as one’s interest lies in logical concepts in the fundamental sense in which they are not dependent upon this or that language, they are not amenable to proper definitions. It is extremely unlikely, of course, that Russell would have entertained anything like (*) as a definition of a logical constant, but the underlying idea that logical constants are too fundamental or too basic to our thought to allow for informative characterizations accords quite well with his universalism about logic plus the view that logic in the fundamental sense is not a matter of language. Two problems remain with the passage from §10 of the Principles, 536 Ch. 5 Logic as the Universal Science II however. Firstly, it does not seem to be about particular logical constants but the generic notion of logical constant or the class of logical constants, as Russell puts it. Secondly, the reason that he gives for thinking that this notion or class should be indefinable appears less than conclusive. What he says is that the generic notion cannot be defined, because some among the logical constants are so fundamental that they would appear in any attempted definition. He does not make clear, however, why this should be a problem. A definition of the generic notion of logical constant with a clause that contains this or that particular logical constant would only be circular if grasp of that constant presupposed an antecedent grasp of the generic notion. Russell gives no reason why this should be so, and the suggestion is in itself rather implausible: in order to grasp (the semantic content of) this or that particular logical constant, one hardly needs to grasp the fact that the constant in question is a logical one.50 The second passage in which the issue of definability surfaces is in §16 of the Principles. Here, though, the point is clearly not about the generic notion but some particular logical constants. What Russell argues is that material implication cannot be defined, because the definition would be circular in a special sense. The passage has already been quoted and discussed, but I shall repeat it here:

A definition of implication is quite impossible. If p implies q, then if p is true q is true, i.e. p’s truth implies q’s truth; also if q is false p is false, i.e. q’s falsehood implies p’s falsehood. Thus truth and falsehood merely give us new implications, not a definition of implication. If p implies q, then both are false or both are true, or p is false and q is true; it is impossible to have q false and p true. In fact, the assertion that q is true or p false turns out to be strictly equivalent to “p implies q”; but as equivalence means mutual implication, this still leaves implication fundamental, and not definable in terms of disjunction.

This passage is about material implication. It is not clear, however, that Russell’s point has anything to do with that notion’s being a logi-

50 As Byrd (1989, p. 348) points out, commenting on the same passage. Ch. 5 Logic as the Universal Science II 537 cal constant. More plausibly, the argument stems from his conception of the canonical form of definitions. Since definitions are universally quantified biconditionals, an attempted definition of material implication only gives new implications:

(Imp) For all p and q, p implies q if and only if not-p or q; that is,

(Imp´) For all p and q, p’s implying q implies and is implied by ‘not-p or q’.

The problem with the suggested definition is not that it is circular in the standard sense of having the definiendum contained in the definiens; (Imp) is not circular in that sense. For a universalist like Russell, however, (Imp) could not accomplish what it is supposed to accomplish. It would be pointless to introduce material implication with the help of a definition, because the definition, assuming the form of material implication, presupposes that notion to be independently available. This is a perfectly reasonable argument to make, when it is seen from the universalist’s point of view; that is, when it is seen as being concerned with material implication in the fundamental sense. And it is evidently more reasonable to construe it in this way than to attribute it to a failure to distinguish between object-language and metalanguage.51 This argument, however, has nothing in particular to do with the fact that material implication is a logical constant; definitions are universally quantified biconditionals, and it is for this reason that material implication (and generality) cannot be defined.52 In this sense, while the two logical constants are indefinable, others can be defined.

51 This view is expressed in Grattan-Guinness (1977, pp. 112-3). 52 Since definitions are universally quantified biconditionals, Russell’s argument can be repeated for the logical constant generality; cf. Byrd (1989, pp. 347-8). 538 Ch. 5 Logic as the Universal Science II

For instance, negation is defined with help of the property that a false proposition implies every proposition.53 Comparing §§10 and 16, we can see that in one of them Russell argues that logical constants or, possibly, the generic notion of logical constant, cannot be defined, while in the other he argues for the indefinability of only some logical constants – two of them, to be precise. We can reconcile the passages by noting that Russell uses “definability” in two senses. The first passage is about logical constants (or the notion of logical constant) and their definability in the philosophical sense of definition, whereas the second passage has to do with definitions in the technical sense.54 In the latter sense, he explains, many logical constants are definable, for they can be singled out by indicating some unique property (by indicating that a logical constant is the unique satisfier of some propositional function or the only term that stands in some relation to some given term). But some logical constants must be recognized as indefinable even in this sense, because they are used in the formulation of such technical definitions. In this argument for the indefinability, the notion of logical constant plays no role, even though the entities to which the argument applies happen to be logical constants: it is strictly about definability in Rus- sell’s technical sense. As regards “definition” in the philosophical sense, he holds the generic notion of logical constant to be indefinable (though the reason given in the text for this view remains somewhat obscure). However, supposing that some notion of philosophical definition is available such that the generic notion of logical constant turns out to be indefinable in that sense, it does not yet follow that the early Russell did not have available to himself a general criterion for logicality, which brings out their distinctive role and explains their philosophical importance, even if that criterion does not qualify as a definition in some strict sense.

53 See §19 of the Principles. 54 See, again, §19 of the Principles. We have already met this distinction in section 4.6.4. Ch. 5 Logic as the Universal Science II 539

The implications of the universalist conception for the issue of logicality are not purely negative. For that conception will of course inform one’s positive considerations concerning the criteria for logicality; this can be seen clearly when we turn to the two notions of logicality mentioned above, the normative notion and logicality as non-particularity.

5.5 Universality and the Normative Conception of Logicality

What does it mean to say that logic is universal, or that logic is a universal science? It is a natural assumption that an answer to this question should draw on some notion of generality. Generality, however, can be understood in different ways. In particular, the normative notion of logicality and logicality as non-particularity can both be explained in terms of generality. Thus we have one notion of generality that is tied to normativity and another notion that is quantificational or purely descriptive. In this section I will consider the normative conception of logicality. We shall see that this conception comes very naturally to the universalist, particularly if he is anti-Kantian. I shall use Frege to elaborate on this claim. Frege’s case is particularly useful in that it helps us to see how the important notion of constitutive rule can be used to illuminate the normative conception. Nevertheless, I will argue that in Russell’s conception of logic it plays no important role. On one understanding of “general”, the claim that logic is a universal science amounts to the view that it has a legislative function for all thought. Kant’s notion of “pure general logic” is a variant of this conception: it is the sort of logic that “contains the absolutely necessary rules of thought without which there can be no employment whatsoever of the understanding” (A52/B76). In this passage “thought” is to be understood in the way Kant understands it, that is, as a term that governs all acts of concept use, like judging, asserting, supposing, 540 Ch. 5 Logic as the Universal Science II inferring, etc. It is important to be clear about what is and what is not involved in Kant’s talk of “rules of thought”. This term, like the traditional “laws of thought” can be taken in either of the following two ways:

(1) logic is normative for all thought: its laws and rules prescribe how one ought to think;

(2) logic is descriptive for all thought: its laws describe how people in fact think.

The quotation from Kant does not disambiguate between these two alternatives, and the qualification of “rules” by “necessary” could well be taken to mean (2). As we saw in section 5.2.2, however, it is nevertheless the normative conception that is relevant here. Kant’s view is thus not that logic should provide a description of factual thinking processes. Its rules and laws are therefore not necessary in the sense of laws of nature; if they were, there could not be illogical thought, as there are no natural processes violating the laws of nature. And even though some philosophers have, indeed, denied the possibility of illogical thought, the standard view, and one that better conforms with common sense, is that people can, and do, make logical mistakes. The use of “necessity” is thus not meant to convey the idea that we cannot but think in accordance with the laws of logic; the point is the more subtle one that the rules and laws of logic are necessary in the constitutive sense. The notion of a constitutive rule is a familiar one. Something that we may refer to as a “practice” – P for short – is created or made possible by laying down a series of conditions or rules, R1, R2,..., Rn; taken together, they serve to define P. A statement “P(Rn)” to the effect that a rule Rn characterises P is an analytic truth, because it supplies a partial definition of P. For instance, chess is partially defined by the rule that the king may move to any adjoining square not attacked by one or more of the opponent’s pieces and not occupied Ch. 5 Logic as the Universal Science II 541 by a piece of the same colour. Hence it is an analytical truth that in chess the king may move in this particular way. “P(Rn)” therefore expresses a conceptual necessity. It follows that there is a particular kind of non-contingent relationship between being a participant in P and the various rules, R1, R2,..., Rn: the acts of someone who is regarded as participating in P are automatically judgeable as correct or incorrect in the light of the relevant rules. Thus, I have made an incorrect (that is, illegal) move, if in the course of playing a game of chess I move my king to a square which is attacked by one of my opponent’s pawns. The mere presence of constitutive rules is not enough to make P a normative practice. Playing a game of chess, for instance, is not a normative practice, even though the rules that define the game are normative in the sense just explained. For this to follow, it must be assumed further that the point of the practice is itself inherently normative. Hence, what makes logic a normative discipline (supposing it is one) is that its point is to supply (some of the) rules of correct thinking (concept-use, etc.) Another important point is that normativity does not exclude description. In the case of logic (and no doubt more generally, too), one naturally asks, what is the source of the relevant rules. Or, what is the ground of the normative practice? And someone who feels the need to explain this may very well think that here something factual is needed, a description of which constitutes the ultimate explanation of the characteristically logical norms. A clear example of this explanatory strategy is found in Frege’s conception of the laws of logic. Frege is willing to admit that the term “laws of thought”, when correctly understood, can be applied to logic. He argues, however, that there are two reasons why this is apt to cause confusion. Firstly, he insists, somewhat in the manner of Kant, that we must be clear about the distinction between normative and descriptive laws.55 When the term “laws of thought” is applied to logic, the only legitimate construal is the normative one; in this sense,

55 See the Introduction to Grundgesetze. 542 Ch. 5 Logic as the Universal Science II the laws of logic prescribe how one ought to think. The alternative would be to think of them as laws describing actual thought processes, but that would lead to relativism and psychologism. The second threat that Frege wants to see off concerns the point of logic. If one fails to distinguish between “true” and “being-taken-to-be-true”, one may well end up thinking that the underpinning for the normative laws is provided by “general agreements among the subjects who judge”.56 This is just confusion, however, based on a failure to abide by the sense of true. For what is true is one thing and what is taken to be true is another thing, and the second is not part of the sense of the first; if it were, it would be a contradiction to maintain that there is something true that everyone takes to be false. But that is no contradiction. Frege is thus very clear about the point behind the normative practice of logic. The source of the laws and rules of logic is not given by what is generally agreed upon, but by the concept of truth, which constitutes the aim of thought (judging, asserting, inferring, etc.). Frege points out that not all laws of truth belong to logic. All sciences have truth as their goal, and all truths are, in this sense, laws of thought. For every descriptive law implies or gives rise to a prescrip- tion that one ought to think in accordance with that law, if one is to attain the truth.57 For example, if one wishes to think correctly about objects with masses, one’s thoughts should not contravene the in- verse square law; the law thus provides a factual basis for a norm of thought. Logic has a special claim to the title “laws of thought” only in the sense that its laws are more general than the laws of physics, for example, and apply whenever we think; they are, indeed, the most general laws of truth. Hence, also, the normative laws that derive from the descriptive laws of logic are laws which “prescribe universally the way in which one ought to think if one is to think at all” (1893, p. 12 (xv)). It is this last phrase, “if one is to think at all”, that provides the

56 Frege (1893, p. 13 (xvi); this phrase is a quotation from Benno Erd- mann, a typical psychologistic logician, according to Frege. 57 Frege (1893, p. 12 (xv)); (1918-19a, p. 351). Ch. 5 Logic as the Universal Science II 543 link between Frege and the notion of constitutive rule. Since he held the view that all thought or all concept use assumes the form of judgment, the Fregean version of the constitutivity idea can be formulated by saying that the most general laws of truth are constitutive of the notion of judging subject.58 According to Frege’s conception of judgment, judgment is the acknowledgment of the truth of a thought.59 Hence, to be a judging subject is to have truth as the goal of one’s use of concepts and for others to make this attribution, i.e., for others to construe one’s acts as acts of asserting, supposing, inferring, etc. And this in turn means that one’s acts are judged as correct or incorrect in the light of the most general laws of truth (the laws of logic). The special, constitutive role of logic underlies an argument for the purely logical character of arithmetic that Frege gives in §14 of the Grundlagen. He points out, first, that “for purposes of conceptual thought” we can always deny an axiom of geometry without involving ourselves in contradictions. (Here “geometry” means Euclidean geometry.) This shows, among other things, that the truths of geometry are non-logical. He then asks whether the same can be said of the “fundamental propositions of the science of number”. The answer to this question is “no”. For if we as much as try to deny one of these propositions, we end up in complete confusion, where thought no longer seems possible (what Frege means by “denying a proposition” is thus “to think of its negation”). This impossibility shows, or at least suggests, that the basic laws of arithmetic should be intimately connected with the laws of thought. Although the conclusion of the argument is explicitly directed against Kant, the framework within which it is formulated bears a close resemblance to Kant’s talk about the “conditions of objective thought”. Above all, Frege and Kant agree that cognition is to be understood as resulting from interplay between sensibility and under-

58 This at any rate is a reasonable interpretation of Frege’s views on logic and logical knowledge; see Burge (1998b) for an extensive discussion. 59 See, for example, Frege (1918-19a, pp. 355-6). 544 Ch. 5 Logic as the Universal Science II standing, both of which supply their own characteristic conditions of knowledge or of objective thought.60 Recall that, according to Kant, the norms supplied by pure general logic” are necessary in the sense that they apply whenever we think, whereas the conditions belonging to the various special logics are contingent, or conditional upon our thinking about this or that particular subject-matter. Frege’s formulations in Grundlagen can be given a similar interpretation. What he says is that the laws of thought (and the laws of number) govern “everything thinkable”. Although this formulation is de re, the point is just that whenever we think (use concepts, judge, assert), and hence whatever we think, the correctness or otherwise of our acts is always decided in part by the “laws of thought”. The laws of logic (and of arithmetic) can therefore also be described as truths governing the widest domain of all; Euclidean geometry, by contrast, governs a domain that is less than all-inclusive, as it supplies rules which apply whenever we make judgments about spatial objects.61

60 In Grundlagen Frege calls “reason” that which takes the place of Kant’s “understanding”; in some of his later writings the connection that reason bears to logic is made explicit by calling it the “logical source of knowledge”. For the term “reason” (Vernunft), see §§26 and 105 of Grundlagen; Frege’s discussion of the sources of knowledge is found in a late manuscript entitled “Erkenntnisquellen der Mathematik und der mathematischen Naturwissen- schaften” (Frege 1924-5). Sensibility in turn is intimately connected with geometry, and hence it is appropriate to refer to sensibility as the “geometrical source of knowledge”; in the Grundlagen, though, Frege simply refers to it as “the intuitable” or “spatially intuitable”. 61 Even though Kant and Ferege shared a common framework, this does not yet decide the issue whether, and if so, in what sense, Frege was a Kant- ian philosopher. This issue is commonly formulated as one concerning the sense that “objectivity” had for them (see, for example, Sluga (1976)), or, more informatively, what they held the objectivity of judgments to consist in. What is essential to Kantianism (more generally, to idealism of any sort) is the explanation of objectivity by recourse to some notion of practice or activity, broadly construed; cf. Burge (1997, p. 6). A typical idealist or Kantian or anti-realist reading of Frege has it that his pronouncements on the status Ch. 5 Logic as the Universal Science II 545

Though the framework is shared, the point of Frege’s argument is directed against Kant. While Kant had thought that very little is left over when the conditions of sensible thought are abstracted away, the logicist Frege believes he can prove that logic or conceptual thought alone is capable of establishing truths that are commonly regarded as substantial, rather than trivial, namely the truths of arithmetic. It is of thoughts, logical objects and other non-spatiotemporal objects should not be taken in a straightforward ontological sense but ought to be seen as containing an implicit qualification. For example, Michael Friedman (1992b, 536) argues that the objectivity of logic for Frege does not consist in the fact that it is concerned with a platonic realm of mathematical objects, but in the fact that the rules of logic make the notion of genuine judgment possible in the first place (this reading of Frege follows Ricketts (1986)). Skipping all the details, I would be inclined to argue that Friedman has misconstrued the order of explanation here. The objectivity of logic for Frege cannot consist in the fact that its rules make the notion of objective judgment possible. In order to do that, these rules must be of a certain kind. Above all, since judgment is the acknowledgement of truth and truth is independent of judging subject, the laws must be construed in a way that complies with this independence. In other words, the constitutive function of logic presupposes objectivity-qua-independence, i.e., a platonistic notion of objectivity, on some suitable, presumably rather minimal, understanding of platonism. Those who argue for a Kantian interpretation of Frege, often refer to §26 of Grundlagen. There the explanation is found that what is objective is independent of sensation, intuition and imagination, but not of reason, for, Frege asks, “what are things independent of the reason? To answer that would be as much as to judge without judging”. That things are not independent of reason could be understood in some suitable Kantian manner (as is suggested by Haa- paranta (1996, sec. 3)). For Frege, however, “reason” seems to mean no more than what is contained in the notion of “most general laws of truth”. The suggested reading is therefore hardly mandatory. Incidentally, Frege’s argumentation in the passage seems uncharacteristically sloppy. The fact that we cannot answer the question, What is x? without making a judgment and, in that sense, relying on reason, scarcely shows that x itself should be dependent upon reason. Perhaps, though, the argument is less sloppy, if we take “reason” to refer to the objective laws of truth. That would be one reason to resist the Kantian interpretation, but this question should not be pursued here any further. 546 Ch. 5 Logic as the Universal Science II thus easy to see why the “conditions of thought” framework should be attractive to a logicist; once one enriches the stock of logical forms that is available in the analysis of the conditions of objective thought, one is forced to reconsider the interrelations of pure thought and sensibility. Summing up the discussion so far, Frege held logic to be a normative discipline in the sense that the most general laws of truth supply criteria that apply whenever the correctness of judgments is at issue. To put the point another way, it is only in the light of these rules that someone can be regarded as a “judging subject” (the constitutive role of logic). What does this imply for the topic of this section, i.e., the suggestion that the universality of logic could be understood in terms of its normativity, or the idea that logic is legislative for all thought? In section 5.2.2 I explored the suggestion by MacFarlane (2000, 2002) that for Kant, formality/logicality means normativity + universality. On this view, “pure general logic” provides the norms that apply whenever thought occurs or concepts are used. Whatever further characteristics apply to this kind of logic, they are substantive consequences of its generality in the normative sense. In Frege’s case the situation is more complicated, mainly because he is clearer than Kant ever was about the source of the normativity of formal logic. According to Frege, logic has normative force, because, firstly, the laws of logic are the most general laws of truth, and, secondly, the concept of truth defines the point behind the practice of judging. It seems, then, that Frege needs not just one but two notions of generality: in addition to generality in the normative sense, there is that sort of generality that pertains to the laws of logic in the descriptive sense. The first sense can be accounted for by drawing on the idea of constitutive rules. The second sense is explained by explaining what makes the content of a logical law or proposition general in a way that is distinctively logical and fits the universalist conception of logic. Frege’s case demonstrates very clearly that the normative and descriptive senses of generality are not exclusive of each other. In cases Ch. 5 Logic as the Universal Science II 547 where both senses are present, the question of logicality comes down to this: how are the two senses, the normative and the descriptive one, related to each other? Is one of them more basic than the other, or are their mutual relations more complicated than that? Fortunately, we need not resolve this interpretative problem here; basically because the normative conception plays a very marginal positive role in Russell’s understanding of what is involved in logicality/generality. It is nevertheless worthwhile to discuss the matter briefly, to bring out some issues that will be discussed in more detail later. For Frege, logicism is an answer to an epistemic question: what are the “springs of knowledge upon which this science [sc. arithmetic] thrives” (1897, p. 235). The point behind the thesis that arithmetic is reducible to logic is to identify the source of our knowledge of arithmetic, and the point of saying that it is reducible to logic is to say that this source is to be found in the laws of thought, rather than in some apriori intuition. That Frege’s version of logicism is primarily epistemic is not in itself enough to resolve the question about the relation between the two notions of generality/logicality. For to hold that the source of arithmetical knowledge is logic and not an apriori intuition is to be committed not only to the epistemic thesis that the ground of arithmetic is pure thought, or laws that cannot be renounced without renouncing thought, but also to a further thesis that the ground consists in laws that are universally applicable, rather than limited to some restricted domain. And this thesis has implications for what kinds of concepts may and may not occur in the propositions of pure logic. The second thesis is very much in the spirit of the Bolzanian account of logic and its approach to logicality; a truth is logical if its expression requires no other means than “logical constants”, to which condition is now added the further one that ties logicality to descriptive generality. It is natural to understand Frege’s talk of the most general laws of truth in this way. An example is provided by the following gloss on “most general laws of truth” by Thomas Ricketts:

Maximally general truths are truths that do not mention any particular thing or any particular property; they are truths whose statement does 548 Ch. 5 Logic as the Universal Science II

not require the use of vocabulary belonging to any special science. (1986, p. 60)

We may give the name “pure descriptive generality” to that notion of generality which Ricketts tries to pinpoint in this passage and which he attributes to Frege. Accounts of logicality based on pure descriptive generality have often been criticized for having counterintuitive consequences. A familiar case in point is provided by claims about the cardinality of domain. Since this issue is familiar from secondary literature, I shall consider another formulation of what is essentially the same critical point. Consider the following way of getting at logical structure, one that is probably more Russellian than it is Fregean. Russell referred to it as “abstraction” or, more often, “generalization” (1903a, §8; 1911a, pp. 36-7; 1913, Ch. IX): given a true statement like “London is south of Edinburgh”, consider “( x)( y)( R)(xRy)”, the existential generalization on its non-logical terms, including the predicate. This statement is maximally general in the sense delineated by Ricketts, for it does “not mention any particular thing or any particular property”. For Frege, however, the logicality of logical truths is at least partly a matter of how they can be known, and one may well doubt whether our knowledge that something is related to something by some relation is plausibly attributed a logical source.62 More simply, there is the following difficulty for the suggested criterion of logicality.63 A logical object is presumably one that can be referred to using logical expressions only. But then a statement to the effect that there are objects other than logical objects of some particular kind is one that can be written using logical expressions only and is therefore a maximally general truth in the above sense – a Fregean example would be the statement, “there are objects that are not value-ranges”. But it is evi-

62 This and related points about Frege and logicality are discussed in Heck (2007, sec. 1). 63 See Heck (2007, pp. 36-8) and Kemp (1995, 35). As Kemp points out, the point goes back at least to Frank Ramsey: see Ramsey (1925, p. 167). Ch. 5 Logic as the Universal Science II 549 dently not for logic to decide whether or not there are non-logical objects. At least from Frege’s point of view, then, there seems to be little reason to think that the criterion which operates with “pure generality” could constitute a sufficient condition of logicality – assuming that something like Ricketts’ formulation is a reasonable explication of what is involved in pure generality. On the other hand, it is not evident that the normative notion can stand on its own, either. Inspired by passages like §14 of the Grundla- gen, the Introduction to the Grundgesetze and Frege’s late essay Der Gedanke, some scholars have argued that for Frege the logicality of a law of logic is entirely a function of the special, constitutive role that it has in thought and reasoning. Here are two prominent quotations in which this view is expressed:

[W]hen it comes to the question what makes a primitive law logical, Frege has nothing to say beyond the appeal to Generality in §14 [of the Grund- lagen.] To ask whether a primitive law is logical or nonlogical is simply to ask whether the norms it provides apply to thought as such or only to thought in some particular domain. (MacFarlane 2002, p. 40)

[For Frege] a logical law is identified by its being a presupposition of all thought or reasoning, and hence of all domains of science. Unlike the laws of special sciences, logical laws cannot for instance be supposed false for the sake of argument, for without them we cannot be said to argue. (Kemp 1995, p. 35)

As against such interpretations one may point out passages from Frege where logicality is being discussed and where “logical constants” or their equivalents feature prominently, rather than the idea that logic constitutes the presupposition of all thought.64 That the

64 I shall mention three such passages. The first is from the Introduction to the Grundgesetze. There Frege comments upon Dedekind’s monograph on the concept of number. He duly notes Dedekind’s logicism, but adds that the book hardly contributes to its confirmation, because the expressions “system” and “a thing belongs to a thing”, which Dedekind uses, “are not usual in logic and are not reduced to acknowledged logical notions” (1893, p. 550 Ch. 5 Logic as the Universal Science II laws of logic are presuppositions of thought is undoubtedly part of Frege’s conception of logic. But it does not seem that it was meant to exhaust the content of the notion of logicality. It may cover much of what he has to say about the topic – in particular, if we think of his insistence that the basic laws of logic must be self-evident as another side to the idea of constitutivity – but at the very least one must leave room for some notion of descriptive generality; and perhaps even consider the possibility that the normative dimension of logic should be understood, in the last analysis, as a function of the descriptive content of the laws of logic.65 viii). In the second passage, also from the Grundgesetze, Frege admits that a similar worry attaches to his own Basic Law V, concerning the identity of courses-of-values. Logicians may not have expressly enunciated this principle, which may raise doubt about whether it really is a law of logic; nevertheless, he argues, it is at least tacitly present in their practice, and he says he holds it to be “a law of pure logic” (1893, pp. vii-viii). The third passage is a later one, coming from one of the essays that Frege wrote on the foundations of geometry after his controversy with Hilbert and Korselt (Frege 1906, pp. 337-40). One of the problems taken up in the essay is whether and how the independence of axioms could be proved. Frege’s proposal as to how the problem should be tackled with parallels Bolzano’s approach to logical properties and thus presupposes a distinction of concepts or expressions into those that belong to logic and those that do not: “It is true that in an inference we can replace Charlemagne with Sahara, and the concept king with the concept desert, in so far as this does not alter the truth of the premises. But one may not thus replace the relation of identity by the lying of a point in a plane. Because for identity there hold certain logical laws which as such need not be numbered among the premises [...].” (id., p. 338) 65 On a somewhat different note, a simple test of logicality of the sort used by Frege in §14 of the Grundlagen is likely to yield results that do not accord very well with our intuitive judgments of logicality. Consider again the generalization “( x)( y)( R)(xRy)”. Since this is true in any non- empty domain, it seems like a good candidate for being a principle underlying all thought. To be sure, it is not an absolute presupposition, for it can be coherently denied – the statement that nothing exists, though evidently false, does not appear to be incoherent – but there is very little left of thought or reasoning, if we are not entitled to assume that the domain for our quantifi- Ch. 5 Logic as the Universal Science II 551

It is worth emphasizing that the normative notion of generality is attractive in more than one way. It in the first place, it is attractive to a logicist. If logic is normative for all thought (an assumption shared by Kant and some logicists), and if, furthermore, logic is not epistemically barren in the way suggested by Kant (this is the logicist view), then logicism shows in a very straightforward way that Kant had been wrong about intuition and its relation to pure thought. Furthermore, when the normative notion of logic is unravelled along the lines considered above – within a framework that builds upon the notion of “conditions of thought” – the logicist is in a position to meet a potential objection, advanced by many critics, that his reductive programme in fact involves a change in the way the term “logic” is used. In other words, the normative conception enables the logicist to meet Kant on his own ground. The normative conception of logicality is also attractive from an interpretative point of view. For it can be put to use in an attempt to make sense of the often-expressed idea that universalism about logic is essentially the view that logic underlies all thought or rational discourse. When we turn to Russell’s case, however, we shall meet a version of the universalist conception – and of logicism – in which this view plays very little role. Accordingly, Russell’s understanding of logicality makes use of the descriptive, rather than the normative concept of generality.

5.6 Russell and the Normative Conception of Logicality

The normative conception of logicality is almost never discussed by Russell. When he considers the relation that logic bears to thought, ers is non-empty, or that something “self-identical exists”. But, again, we may ask: should this assumption be attributed to logic? At the very least, then, a defender of the normative notion of logicality needs a more incisive criterion for deciding whether some principle is constitutive of “thought as such”. 552 Ch. 5 Logic as the Universal Science II what is at issue is almost always the question whether the laws of logic are descriptive of thought, when the latter is understood in some psychologically realistic way. As is to be expected, he rejects such views offhand, on the grounds that they lead to psychologistic conceptions of logic. Here are two examples of his dismissive attitude:

[Boole] was [...] mistaken in supposing that he was dealing with the laws of thought: the question how people actually think was quite irrelevant to him, and if his book had really contained the laws of thought, it was curious that no one should ever have thought in such a way before. (1901a, p. 366)66

The name ‘laws of thought’ is also misleading, for what is important is not the fact that we think in accordance with these laws, but the fact that things behave in accordance with them; in other words, the fact that when we think in accordance with them we think truly. (1912a, pp. 40-41; emphasis in the original)

The second quotation is particularly useful in that it indicates Rus- sell’s reasons for dismissing psychologism; although the quotation is from The Problems of Philosophy, there is no reason to think that there should be a difference in this respect between his earlier and later views. Like Frege, he insists on a strong construal of the notion of objectivity, whereupon the objectivity of the objects of knowledge and cognition amounts to their possessing a special ontological status, signalling their independence of cognition.67

66 The book to which Russell refers in this passage is Boole’s An Investiga- tion of the Laws of Thought of 1854. 67 Here “independence” is to be construed sufficiently broadly, so as to include not only “independence of the knowing subject” but various intersubjective notions as well; for Russell’s views on this matter, see, for example, §427 of the Principles. In particular, this construal rules out the possibility of explaining objectivity in terms of intersubjectivity: when Russell explains that “being is a precondition, not a result of a [thing’s] being thought of”, where being is that which belongs to everything, his point must be taken to apply, no matter what content is given to “being thought of”. Ch. 5 Logic as the Universal Science II 553

Something’s being the case is independent of its being taken to be the case. It is therefore indifferent to the content and character of the laws which describe how things are that we think in accordance with them, if we do. And since logic is no less descriptive of reality than the more familiar descriptive laws that we usually call “laws of nature”, this applies to the laws of logic as well. The point can also be formulated using the notion of truth, as in the second quotation above, as there is an intimate connection between truth and something’s being the case.68 When the truths of logic are understood as descriptive of what is the case, they are not “laws of thought”; they do not tell how we in fact think or how we must think. The laws of logic, Russell holds, are laws of things, and their connection with thought is simply that if we think in accordance with these laws, then, to that extent, we think truly. Given this, we can infer that Russell’s views on the issue of normativity, though largely implicit, were probably not that dissimilar from those that Frege had formulated somewhat earlier. That is, logic has normative bearing upon our thought, because to think truly means, among other things, to think in accordance with the truths of logic. In the Problems of Philosophy Russell explains that the validity of an inference depends upon a “general logical principle”, and if anyone is to ask us why he or she should accept a particular inference as valid, we can only respond by appealing to the principle or principles underlying the inference in question (1912a, p. 40). To accept that logic has normative force is not yet to commit oneself to the view that logic should be given a constitutive role in our thought. So far as I have been able to determine, there is only one passage in which Russell explicitly considers something like this idea. More precisely, in the passage the suggestion is made that the only way to render the normative and descriptive senses of “law of thought” compatible with each other is to regard the laws as constitutive of “mind” in some idealized sense. The passage is an early one, coming from Russell’s first published essay, a review of Die Gesetze

68 See section 4.5.7.2 for the early Russell’s construal of this connection. 554 Ch. 5 Logic as the Universal Science II und Elemente des wissenschaftlichen Denkens by G. Heymans (Russell 1895). What is under discussion in this passage is Heymans’ definition of epistemology, which runs as follows: “The exact determination and explanation, by means of the empirical investigation of given thought, of the causal relations which condition the occurrence of conviction in consciousness”.69 Russell writes:

In order to get a theory of knowledge out of psychology, it has been necessary to leave out of account the various emotional and pathological causes of belief, which certainly come under the above definition; it is throughout assumed that no causes of belief are possible except the evidence of the senses, logical reasoning, and certain fundamental à priori laws, i.e., the grounds of correct belief. In short the psychology of belief appears to be rather that which would apply to an intellectually ideal man than to men liable to error; accordingly error is regarded as springing only from deficient analysis, and this is the only answer we obtain to the question how the laws of thought can be at the same time norms for correct thinking and psychological laws for all thinking. (id., p. 252; emphasis in the original)

The review was written during Russell’s idealist period, but the criticisms of psychologism that are expressed in it were something that he could incorporate into his realism without any substantial modifications. The same applies, arguably, to the points that he makes about the “laws of thoughts”, though the wording may be less than wholly felicitous from the later Russell’s perspective. To study the psychology – perhaps even the Mind – of an “intellectually ideal man” means no more and no less than to discover and formulate principles of correct thought.70 Applied to us – agents who are intellectually less than ideal – these principles assume the form of

69 Translated by Russell (1895, p. 251). 70 Frege suggests a similar formulation towards the end of Der Gedanke, where he writes that although logic and mathematics have nothing to do with human psychology, their task could perhaps be described as “the investigation of the mind [die Erforschung des Geistes]; of the mind, not of minds ” (1918-19a, p. 369). Ch. 5 Logic as the Universal Science II 555 norms; they supply the most fundamental of those standards in the light of which our thought is judged as correct or incorrect. Evi- dently, such principles have nothing to do with empirical psychology. As in Frege’s case, Russell’s point is probably best formulated with the help of the concept of constitutive rule; the principles describing the psychology of an “intellectually ideal man” are ones that help to define what it is to be a thinking and judging subject – and it is for this reason that they are normative for us, i.e., they are normative for us insofar as we act as thinking and judging subjects. Such a view would certainly be compatible with the logicist Russell’s views on logic. Whether it really was his view, however, is doubtful. Note, to begin with, that this conception of the laws of logic is a contribution to the epistemology of logic; that logic is constitutive of thought is a view that could be introduced to explain why we are justified in accepting a law of logic – this is how Frege construes it.71 For the early Russell, however, the epistemic issue of justification was of little concern. It is therefore quite understandable that what he has to say about the laws of logic focuses on the laws themselves, rather than our grasp of them. Hence, if the view under discussion was Rus- sell’s, it was one that is there only implicitly. Secondly, it must be admitted that some of the logicist Russell’s remarks on logic could be interpreted along the lines suggested. As has already pointed out, the universalist conception of logic is often connected with the view that logic constitutes the ultimate presuppositions of thought, and to hold this view is to attribute a constitutive role to the laws of logic (at any rate, this is a reasonable way to under-

71 Cf. Frege (1893, p. 15), where he explains that “[a]nyone who has once acknowledged a law of truth has by the same token acknowledged a law that prescribes the way in which one ought to judge, no matter where, or when, or by whom the judgment is made”. Frege refuses to speculate on the grounds of this acknowledgement, admitting that our acknowledging a law of truth is perfectly compatible with our supposing that someone might reject it; but it does not leave room for doubt as to who is right, we or those who reject the law. In Frege’s view, then, the constitutive role of the “laws of truth” is a reason for taking them to be true 556 Ch. 5 Logic as the Universal Science II stand the talk about “ultimate presuppositions”):

If a principle P is a presupposition of thought, one cannot relinquish P without ceasing to qualify as a thinking subject.72

It could be argued that this principle underlies Russell’s claim about independence proofs that is found in §17 of the Principles (cf. section 4.5.4). His view, it will be recalled, was that the axioms of a logical calculus cannot be assumed false, because these axioms are principles of deduction, and if they are false, reasoning becomes impossible, i.e., we can no longer draw correct consequences, or establish what really follows, from our premises or assumptions.73 Though possible, this reading is certainly not the only possible one. Russell’s point was just formulated using the term “impossible”. In fact, what he literally says is no more than that “all our axioms are principles of deduction; and if they are true, the consequences which appear to follow from the employment of an opposite principle do not really follow, so that arguments from the falsity of an axiom are here subject to special fallacies”. On the face of it, this is a considerably weaker formulation than what is required before we can legitimately infer the above principle from the passage. For the claim is not really that reasoning becomes impossible, if a principle of deduction is assumed false; what he says is that if this assumption is made, then the subsequent deductions that are effected on the basis of this assumption are subject to special fallacies. Other similar-sounding passages can be found from Russell.74 In them the point is invariably made that the axiom or principles underlying deduction must be true, or absolutely true, for otherwise there would not be any true consequences. Admittedly, on occasion he uses stronger wording than what is found in the section from the Principles. Thus, in the Problems of Philosophy he argues that some self-evident

72 Cf. here section 2.7. 73 Kemp (1995, p. 35) attributes this view to both Frege and Russell. 74 See Russell (1911a, p. 37; 1912a, p. 40; 1919, p. 191). Ch. 5 Logic as the Universal Science II 557 logical principles must be admitted before any argument or proof becomes possible (1912, p. 40). Nevertheless, to attribute to him the idea of constitutive rules on the basis of this textual evidence is to make a rather strong conjecture. And it is not clear that room can be made for any notion of constitutive rule in the early Russell’s philosophy (cf. here section 5.10). My general conclusion is that the normative notion of generality does not play any important role in Russell’s conception of logic – one that is similar to the role that it seems to play in Frege’s logicism. The gist of the early Russell’s account of logicality is to be found in a quantificational or purely descriptive notion of generality.

5.7 Russell and the Descriptive Account of Generality

5.7.1 The Development of Russell’s views on Generality

The gist of the early Russell’s account of logicality is to be found in a purely descriptive notion of generality (quantificational generality). Recall how this point of view emerged within the context of Russell’s philosophy. The starting-point is the Bolzanian or purely schematic notion of logical formality. On this approach, the so-called formal properties of a proposition are those that it has in virtue of exhibiting a particular form. This in turn is accounted for by dividing the constituents of the proposition into constants – those constituents whose presence in a proposition gives it its form – and those that are subject to variation. Since the distinction is drawn in a way that imposes no initial restrictions on what constituents may be chosen as constants, it remains purely schematic and must be completed by a suitable criterion for distinguishing between the two kinds of constituents. Since an advocate of the Bolzanian account seeks to explain the distinctively logical properties of propositions – he is interested in the distinctively logical formality – this should naturally be reflected in the criterion that is eventually proposed. In this way we arrive at the classical 558 Ch. 5 Logic as the Universal Science II problem of logical constants: in virtue of which property or properties do logical constants differ from other constants? We can also formulate the question in the following way. The propositions of logic are those in which, to use Quine’s phrase, only logical constants occur essentially.75 How do we distinguish these propositions from the non- logical ones – apart from resorting to the bluntly circular characterization that makes use of the concept of logical constant? We have already seen – cf. section 5.4 – that the early Russell was sceptical about the possibility of framing definitions of logical constants, when definition is understood in some suitable philosophical sense. Imposing sufficiently strict conditions on definitions, then, his view is that logical constants cannot be defined except by enumerating them. This does not mean, however, that he did not have a general conception of logicality or that he would have been inclined, in the manner of some other advocates of the Bolzanian account, towards a largely pragmatic notion of logical constant; the philosophical use to which he wished to put logicism would have rule out that option, had he ever considered it. The early Russell’s conception of logicality is one in which a fairly simple quantificational account of generality occupies a central role. One way or another, some notion of generality is likely to figure in every conception of logic; for logic is concerned with correct reasoning, and it is a natural assumption that there are, indeed, principles of correct reasoning that are general in the sense that they apply everywhere; this is the idea of topic-neutrality. As we have seen, even Kant found room for topic-neutral principles, even if he also thought that there was very little one could do with them by way of positive advancement of knowledge. On the model-theoretic conception, this generality is secured by quantifying over models (interpretations). As

75 Quine (1936, pp. 324-5); essentially the same idea is discussed in Quine (1970, Ch. IV). As Quine uses the term, “essential occurrence” is just an alternative way of saying that some constituents of propositions – for Quine, sentences – remain constant while the others are subject to systematic variation. Ch. 5 Logic as the Universal Science II 559 we have seen, Russell had more than an inkling of the notion of interpretation; after all, the idea that a mathematical theory has a plurality of applications and that the theory itself, considered apart from its applications, is a matter of logical structure was an essential part of the conception of mathematical theories that he accepted. Trying to incorporate this standpoint into his conception of logic, however, he resorted to quantification over ordinary entities, rather than some more exotic ones. In spite of the relative simplicity of the idea, we can distinguish two stages in Russell’s thinking about logical generality. Consider, to begin with, the following two quotations. The first is from “Recent work on the Foundations of mathematics”:

Logic, broadly speaking, is distinguished by the fact that its propositions can be put into a form in which they apply to anything whatever. (1901a, p. 367)

The second quotation is from the May 1901-draft of the Principles:

And logic may be defined as (1) the study of what can be said of everything, i.e., of the propositions which hold of all entities, together with (2) the study of the constants which occur in true propositions concerning everything. (1901d, p. 187; italics in the original)

Let us put aside for the moment the second part of this characterization – “the study of the constants...”. Formulating the point in terms of truth, Russell’s idea is apparently that logical truth equates with general truth, where “general” is meant to be understood in the strict sense of absolute or unrestricted generality; to be logically true means to be true about absolutely everything there is, and logicality is thus captured by the mere presence of universal quantification, provided it really is universal, rather than something else. It is not difficult to see that this part of Russell’s characterization of logic – “the study of what can be said of everything” – is based on what from our perspective is a non-standard construal of quantifica- 560 Ch. 5 Logic as the Universal Science II tion. Understood in the usual way, a universally quantified statement of the form “every F is G” is really a conditional, and can therefore be said to be about everything that is in the range of the quantifier- phrase “every”; and if that includes unrestrictedly everything, we can presumably say that the statement is about absolutely everything. Speaking in the standard way, then, a statement like “all ducks waddle” is about everything, both ducks and non-ducks, but it clearly does not belong to logic. Russell’s non-standard construction presupposes his theory of denoting concepts. On Russell’s first version of the theory, a denoting concept like /all as/ is characterized by the following two clauses, which, when expanded upon, explain the theoretical utility of denoting concepts. Consider, for example, the denoting concept /all ducks/; this concept

(i) denotes those and only those entities that are ducks; (ii) is about those and only those entities that are ducks.76

More generally, Russell holds that a denoting concept denotes – and is therefore about – those and only those entities which are members of the class of the concept that occurs as a constituent in the denoting concept. For example, the proposition /all ducks waddle/ is about each and every duck. This is so in virtue of the proposition’s containing as a constituent the denoting concept /all ducks/, which denotes the class of ducks (I omit the finer details about the denotation relation, since they are irrelevant here: see section 4.4.9.3 for some of these). When denoting concepts are understood in this way, their aboutness depends upon the class-concept occurring as a con-

76 Recall from section 4.4.9.1 that the theory of denoting is introduced to explain “aboutness” in those cases where the entities that a proposition is about cannot plausibly be taken to be constituents of the proposition. This motivates the straightforward connection between denoting and aboutness; the former does not reduce to the latter, however, for not all cases of the former are cases of the latter. Ch. 5 Logic as the Universal Science II 561 stituent of the denoting concept. Since classes of entities typically have fewer than all entities as their members, quantifiers construed as denoting concepts typically range over less than everything. It is only when the class that is denoted by the concept includes everything that the proposition denotes, and therefore is about, everything. Since “term” is Russell’s word for any entity, it is natural to fix upon /all terms/ – or /every term/ or /any term/ – as the denoting concept that is characteristic of the propositions of logic. It follows from all this that the propositions of logic can be characterized as those propositions that are about all terms, where this “aboutness” is to be understood in the manner just explained. This account of the propositions of logic is attractive mainly because of its apparent simplicity – ignoring the complications involved in the notion of denoting. Simple or complex, the theory did not survive into the published version of the Principles.77 For Russell came to reject the account of generality that we have just considered. Instead of holding that a general proposition containing denoting concept denotes and is about those entities only that are members of the relevant class, he came to hold that the variables occurring in a general proposition are unrestricted even in those cases where the proposition is an ordinary, non-logical proposition. Russell’s reasons for this change of mind may be spelled out as follows:

[1] Consider a sample general proposition expressed in the style of Peano, i.e.,with the help of propositional functions, for example “x is human implies, for all values of x, x is mortal”; call this (*). [2] Appearances notwithstanding, (*) does not express a relation between two propositional function; for only propositions stand in the relation of implication. [3] Rather, (*) asserts a class of implications, namely all those im-

77 Accordingly, the passage quoted above, Russell (1901d, p. 187), was deleted from the final manuscript of the Principles, which was completed in May 1902. 562 Ch. 5 Logic as the Universal Science II

plications that result from substituting a constant for the variable. [4] Though convenient for the technical development of formal logic, propositional functions are philosophically to be understood with the help of denoting concepts. [5] Suppose, in accordance with Russell’s first version of the theory of denoting, that /every a/ denotes only as. [6] Combining [2] and [5], we get the result that “x is human implies, for all values of x, x is mortal” amounts to the simulta- neous assertion of as many propositions as there are members in the class of human beings. [7] Consider some propositional function, say , which is false for all values of x. Given [6], we should have to conclude that a proposition like /x is unicorn implies, for all values of x, x has one horn/ asserts as many propositions as there are members in the class determined by . Since the class has no members, “x is a unicorn implies, for all values of x, x has one horn” asserts no proposition. [8] At least in the context provided by Russell’s conception of proposition, however, this conclusion seems false; “all uni- corns have one horn” seems as much an assertion as, say, “all rhinoceroses have one horn”.

Russell expresses this conclusion in §41 of the Principles: “unless we admit the hypothesis [of an implication] equally in the cases where it is false, we shall find it impossible to deal satisfactorily with the null- class or null propositional functions”. The alternative to [6] is the view which Russell formulates as follows:

We must [...] allow our x, wherever the truth of our formal implication is thereby unimpaired, to take all values without exception; and where any restriction on variability is required, the implication is not to be regarded as formal until the said restriction has been removed by being prefixed as hypothesis. (ibid.) Ch. 5 Logic as the Universal Science II 563

Once this view is adopted, however, the characterization of the propositions of logic with the help of the notion of unrestricted variable is no longer possible.78 The proposition of logic, then, must be singled out by some other characteristic. This brings us to the second part of Russell’s definition of logic, to wit, that it is “the study of the constants which occur in true propositions concerning everything.” This will no longer do as it stands, for the locution “true proposition concerning everything” cannot be used to single out the class of distinctively logical propositions. Of course, the propositions of logic are generalizations about absolutely everything, but that does not distinguish them from any other general propositions.79 The alternative to which Russell resorts is an explicit notion of logical constant – the only constants that occur in the propositions of logic (or of pure mathematics) are logical constants – which comes with a criterion for a constant’s being logical that makes use of the idea of a proposition’s content being general.

5.7.2 Russell’s Account of Logical Generality

The contrast involving generality that is relevant for understanding Russell’s conception of logicality is not between unrestricted and restricted generality – this was his first idea – but between generality and particularity.

78 On this point, cf. Levine (2001, pp. 220-1). 79 A qualification is needed here. Russell says “term” is the most general word of his philosophical vocabulary. This leads one to expect that “term” covers everything. This, however, is not the case. For it covers only everything that is, or can be counted as one (cf. §47 of the Principles). This excludes what Russell calls “classes as many”, classes considered as aggregates, rather than classes as wholes (see §70). These “objects”, are many, rather than one, and are therefore not logical subjects and values of variables, either (for further discussion, see Cocchiarella 1980). It turns out, then, that even for Russell, “absolutely everything” does not include absolutely everything, although it certainly includes absolutely every thing; cf. here Proops (2007, pp. 4-5). 564 Ch. 5 Logic as the Universal Science II

But not just any sort of generality will do here. After all, any science, even if we should call it special or particular, strives for generality and abstracts to that extent from certain particularities.80 Never- theless, generality and generalization are central to Russell’s understanding of logic. This is clearly shown by the following quotation from the Principles:

We can now understand why the constants in mathematics are to be restricted to logical constants in the sense defined above. The process of transforming constants in a proposition into variables leads to what is called generalization, and gives us, as it were, the formal essence of a proposition. Mathematics is interested exclusively in types of propositions; if a proposition p containing only constants be proposed, and for a certain one of its terms we imagine others to be successively substituted, the result will in general be sometimes true and sometimes false. Thus, for example, we have “Socrates is a man”; here we turn Socrates into a variable, and consider “x is a man.” Some hypotheses as to x, for example “x is Greek,” insure the truth of “x is a man”; thus “x is a Greek” implies “x is a man”, and this holds for all values of x. But the statement is not one of pure mathematics, because it depends upon the particular nature of Greek and man. We may, however, vary these too, and obtain: if a and b are classes, and a is contained in b, then “x is an a” implies “x is a b.” Here at last we have a proposition of pure mathematics, containing three variables and the constants class, contained in, and those involved in the notion of formal implication with variables. So long as any term in our proposition can be turned into a variable, our proposition can be generalized; and so long as this is possible, it is the business of pure mathematics to do it. (Russell 1903a, §8; emphasis in the original)81

80 Resorting to the Bolzanian account of formality, we can say that all sciences are “formal” to a varying degree. 81 See Russell (1911a, p. 35) and (1914a, pp. 52-4) for similar passages. The phrase “logical constants in the sense defined above” at the beginning of the quotation refers to §1 of the Principles. Having first defined pure mathematics as the class of formal implications containing only logical constants, he gives the following definition of what he means by “logical constants”: “And logical constants are all notions definable in terms of the following: Implication, the relation of a term to a class of which it is a member, the notion of such that, the notion of relation, and such further notions as Ch. 5 Logic as the Universal Science II 565

As the quotation shows, the disregard for particular content does not mean that logic has no content. That it does have content is shown by the fact that generalization is not only a process which dispenses with constants, but also a process which introduces new constants as well. These constants, furthermore, seem to be perfectly like the constants that occur in propositions belonging to special sciences. A special scientist, we may say, endorses generalizations about some particular kinds and properties; the task of the special scientist, we might say somewhat simplistically, is to establish such claims as, for example “if x and y are gluons, then ...x,y...”, that is, general claims about certain kinds of entities. If this characterization is accepted, and if one follows Russell’s description of generalization, one is thereby committed to the view that a statement like “if a and b are classes, and a is contained in b, then every x which is an a is a b” is about classes. Russell is thus committed to the following two theses:

[UL-1] Like any other science, logic has its own characteristic concepts.

[UL-2] The propositions of logic are in a perfectly straightforward sense about the characteristic concepts of logic (exactly in the way that the propositions of a special science are about its characteristic kinds and properties).

The so-called logical constants turn out to be entities that are in almost every respect on a par with other entities (with other terms, as Russell would have put it). In particular, this means that they are constituents of propositions; as Russell says in the quotation above, the proposition expressed by “if a and b are classes, and a is contained in b, then every x which is an a is a b” contains the constants class, contains in and those that are involved in formal implication. It turns out, then, that the relation of implication, for example, is not a special propositional connective, but a relation that holds between some may be involved in the general notion of propositions of the above form.” 566 Ch. 5 Logic as the Universal Science II terms (namely certain propositions), but not others. When logic is considered as a theory, its function is simply to state as many properties of logical constants as are relevant for the purposes of logic. For example, Russell’s propositional calculus is just a theory that states a number of properties of the relation of implication. The following explanation from the beginning of Theory of Impli- cation is indicative of Russell’s views in general:

[T]he subject to be treated in what follows [...] is in fact the theory of how one proposition can be inferred from another. Now that in order that one proposition may be inferred from another, it is necessary that the two should have that relation which makes the one a consequence of the other. When a proposition q is a consequence of a proposition p, we say that p implies q. Thus deduction depends upon the relation of implication, and every deductive system must contain among its premisses as many of the properties of implication as are necessary to legitimate the ordinary procedure of deduction. (1906d, p. 159)

There is thus a perfectly good sense in which, on Russell’s view, logic is a “material” science. The respect in which logic differs from special sciences is just the respect in which logical constants differ from non- logical constants. And the difference is that logic is more general than special sciences; indeed, it is characterized by maximal generality. Hence, logical constants are the constants that feature in maximally general truths. To repeat, since all sciences thrive for generality, this is the only difference between propositions that are logical-cum-general and those that are general but non-logical. This means also that generality cannot provide an independent characterization of logic. There is therefore no characterization of the propositions of logic available to Russell that would be independent of a fairly straightforward appeal to the notion of logical constant; to recognize a truth of logic, one has to recognize that there are properties with respect to which generalizations can be pursued. It was argued above that generality of content is the hallmark of logicality for Russell. We are now in a position to see, however, that this notion cannot be developed into a criterion of logicality that would not de- Ch. 5 Logic as the Universal Science II 567 pend upon an antecedent provision of constants characteristic of logic. Russell is thus committed to the following thesis as well:

[UL-3] Logicality is essentially a matter of maximal generality; understanding what maximal generality is, however, presupposes that one understands the notion of logical constant, or the distinction between logical and non- logical constants.

These three theses, [UL-1], [UL-2] and [UL-3], constitute the core of Russell’s universalist conception of logic. And this means that they, together with their implications, form what is defensible in the claim that Russell’s was a universalist conception of logic. By including [UL-1] and [UL-2] among the core theses, I want to emphasize the material or content-character of Russell’s logic. For instance, characterizations of logic sometimes include the notion of topic-neutrality. In Russell’s case, this is a straightforward consequence of [UL-3]; logic is a maximally general science, because it is concerned with inference in general, and therefore its characteristic concepts and propositions have applications to every subject-matter. As Russell understands it, however, topic-neutrality does not mean that logic would not have a “topic” of its own; to say that logic is topic-neutral is to say that it applies to any topic – that, incidentally, includes logic, too – and this is a consequence of the special character of the concepts and propositions of logic.

5.8 Logic as Synthetic

In the introduction to this chapter I listed four ingredients of the view that logic is synthetic:

$ Logic does not abstract entirely from the relation of thought to objects. 568 Ch. 5 Logic as the Universal Science II

$ Logic is about the world. $ There is no deep distinction between the form and content of propositions; in particular, the notion of form does not do independent, explanatory work, when it comes to the propositions of logic. $ Logic is capable of providing the foundation for mathematics.

The first of these theses is meant to be taken in a non-controversial sense. For Kant, that logic abstracts from the relation of thought to objects follows from the fact its norms are “absolutely” necessary, that is, they apply, whenever there is thought. Since, however, thought is, in a minimal sense, intelligible apart from reference to objects, there can be such absolutely necessary rules only on the condition that they abstract from the relation to objects (sensibility). Rus- sell’s logic recognizes no such distinction. All deductive reasoning is based on “logical forms”, and therefore logic does not abstract from the relation to objects. This thesis is of course not restricted to Rus- sell, but is built into predicate logic. Though in this way uncontroversial (provided predicate logic is “uncontroversial”), the view is not trivial, either. For the creation of a logic that is capable of expressing contents and reasonings that Kant had assimilated into special logics marks an absolutely crucial difference between Kant and the advocates of the “new logic”. The second thesis, that logic is about the world, is a consequence of the early Russell’s construal of the propositions of logic as maximally generally propositions. Later, in The Problems of Philosophy, he would put the point as follows: what is important about the so-called laws of thought, i.e., the laws of logic, is not that we think in accordance with them, “but the fact that things behave in accordance with them, in other words, the fact that when we think in accordance with them we think truly” (1912a, pp. 40-1). When it comes to their content, the propositions of logic are just like any other (scientific) propositions: they state general truths about the structure of the world, not, as Kant would have it, about our thought about the Ch. 5 Logic as the Universal Science II 569 world. The third thesis captures the idea that logic itself is substantive and has its own ontology, in addition to being about the world. This point can be formulated in both metaphysical and logical terms. Rus- sell’s philosophy recognizes no Kantian distinction between thought as such and object-related thought; for Russell, all thought is object- related, as thought is simply a matter of apprehending or grasping propositions which are out there. The relation that the thinking subject bears to logic is in the last analysis no different from the relation that it bears to any other subject-matter. Like Bolzano and Frege – and, no doubt, others – before him, Russell argued that objectivity can only be understood in ontological terms; the category of terms, or entities which have being, is Russell’s proposal as to how this requirement should be fulfilled.82 That was the metaphysical formulation. The logical formulation brings us to the Bolzanian account of logic and its basic assumption, to wit, that there is no deep or explanatory distinction between form and content. Since logic has ontology, this is just another formulation of the point made in the previous paragraph. The entities that constitute the subject-matter of logic all belong with content. When unpacked, this claim leads to the theses [UL-1] and [UL-2] discussed in the previous section. Finally, there is the fourth claim about syntheticity and content, to wit, the thesis of logicism. I have hardly touched upon this question in the present work. It was mentioned in chapter 1 that the question whether logicism really succeeds in reducing mathematics (or some part of mathematics) into logic, rather than something else, is often discussed in the context of the paradoxes. Applied to Russell, the idea is the familiar one that he succeeded in maintaining logicism only at the cost of introducing evidently non-logical principles into the reductive base, the axiom of reducibility and the axiom of infinity being the prime examples here.83 To this we may add that this move was

82 Cf. here §427 of the Principles, and see the discussion in section 4.4.1. 83 Russell himself expressed such views: see Whitehead and Russell 570 Ch. 5 Logic as the Universal Science II accompanied by another, to wit, a substantial change in the way logic is conceived. Remember, again from chapter 1, the idea, perhaps attributable to Poincaré, that logicism in fact amounts to no more than stipulation. On this view the logicist pretensions are in the end no different from declaring that the reductive base belongs to the sphere of “logic”. Now, one of the more intriguing views about logic that the post-paradox Russell endorsed was the epistemology that he formulated in his (1907b). According to this epistemology, our reasons for believing the propositions in the reductive base are not such as could be characterized in typical rationalistic terms (self-evidence, etc.). They should rather be seen as broadly inductive: the reason why we trust the base theory is that we have good reasons to believe its consequences. If, now, this view is combined with logicism, the result is what has been called conjectural logic.84 Such a view has little to do with any of the characteristics that tradition has attached to logic (this, in a way, was just Poincaré’s point), and although the new logic evidently presupposes some amount of redescription, there presumably is some limit to what is and what is not acceptable in this direction. The issues relating to the paradoxes do not belong to the present work, however. But the question raised above about logicality does have an obvious implication for our concerns. For one central difference between Kant and Russell is precisely that they disagree about the ontological consequences of logic. For Kant logic – that is, pure general logic – is without presuppositions,85 while Russell’s claim about the content-character denies this view. Now, one could argue that on this particular issue the paradox tips the scales in favour of Kant. This suggestion has been endorsed by Hylton:

(1910, p. xiv,), Russell (1911b, 52-3). 84 Cf. Coffa (1991, pp. 120-2). 85 Here I ignore the fact that the source of pure general logic is to be found in the understanding: Whether this relationship might be seen as explanatory of logic and if so, how substantive implications it has for the nature of logic is a question for Kantian scholarship. Ch. 5 Logic as the Universal Science II 571

For Kant, logic has no objects of its own, and does not even deal with objects; its concern is with the understanding and its form [...]. Russell’s propositions and propositional functions, by contrast, are logical objects [...]. One way to understand the significance of Russell’s paradox, and related paradoxes, is as showing that Kant was right on this issue, and Rus- sell and Frege wrong. The assumption that there are logical objects, when combined with the generality which both Russell and Frege took as characteristic of logic, leads to paradox. (1990b, p. 214, fn. 39)

Hylton is correct in asserting that the paradoxes impose some limitation on the “unrestricted generality” that is characteristic of Russell’s early logic; as Russell notes in the Principles, §102:

The reason that a contradiction emerges here is that we have taken it as an axiom that any propositional function containing only one variable is equivalent to asserting membership of a class defined by the propositional function.

And when Russell suggests that the contradiction might be evaded by suitably modifying the notion of all terms or of all classes, he is loath to accept such measures for reasons that he clearly thinks have something to do with the very nature of logic:

It might be urged that no such sum-total is conceivable; and if all indicates a whole, our escape from the contradiction requires us to admit this. But we have already abundantly seen that if this view were maintained against any term, all formal truth would be impossible, and Mathematics, whose characteristic is the statement of truths concerning any term, would be abolished at one stroke. (id, §105)

What leads Russell (and Frege) to paradox, Hylton argues, is the assumption that there are logical objects, combined with their characteristic view of logical generality. I would maintain, as against this view, that the culprit here is the notion of generality, and not the view that logic has its own characteristic objects and concepts.86 The view

86 Indeed, the last sentence of the quotation from Hylton seems to admit 572 Ch. 5 Logic as the Universal Science II that logic has ontological consequences can be maintained even after it is recognized that the doctrine of the unrestricted variable cannot be retained in its original purity.87 As the discussion above shows, the content-character is very much independent of the issues relating to logicism. It seems clear enough that a dispute between a Kantian and a Russellian view about logic and its resolution involves a great deal more than just questions about the paradoxes.

5.9 The Demarcation of Logic: Russell on Valid Inference

5.9.1 The Propositions of Logic as Formal Implications

We can deepen our understanding of Russell’s version of the Bolza- nian account of logic by considering what implications it has for his views on inference. The conception of logical generality which was discussed above leads Russell to the view that the propositions of logic are certain kinds of formal implications. As we saw in section 4.4.10.3, a formal implication is almost like a universally quantified conditional, or a statement of the form for all x, F(x) Ⱥ G(x). One qualification is needed, however. It is that a formal implication does not really express a relation between two propositional functions, for only propositions can stand in the relation of implication; hence a formal implication is to be understood as an assertion of a class of implications possessing a particular form. Clearly, not all proposition of this particular form belong to logic. as much; what leads to paradoxes is not just the view that there are logical objects. Russell’s propositional functions, Hylton says, are logical objects. Taken in itself, the notion of propositional function can hardly be responsible for the paradoxes, for some such notion is needed, no matter how thinks of logic. 87 Exactly what happens to the unrestricted variable is one of the key questions about the development of Russell’s logic. It must be said that the question is still very much open; for two contributions, see Landini (1998) and Stevens (2005). Ch. 5 Logic as the Universal Science II 573

For instance, “for all x, if Human(x) Ⱥ Mortal(x)” would not qualify as a logical proposition, because such particular properties as humanity and mortality and their interrelations fall outside the sphere of logic; we might say that in this case the generality characterizing the proposition, is not of the right kind. And yet the proposition is a formal implication (Principles, §12). To qualify as a proposition of logic, the implication must fulfil the further condition that the constants it contains must be logical in character.88 As we have seen, Russell does not invoke a substantial notion of formality to explain what makes the propositions of logic distinctively logical. That this is so is already shown by the fact that he is willing to apply the adjective “formal” to any implication of the form for all x, F(x) Ⱥ G(x). Used in this way, formality in effect reduces to generality. If we follow the Bolzanian account of logicality (cf. section 5.3.2), we can say that “all humans are mortal” is a formal truth with respect to humanity and mortality, meaning that every proposition exhibiting the form “Human(x) Ⱥ Mortal(x)” is true. In the Principles the main point is formulated as follows:

It seems to be the very essence of what may be called a formal truth, and of formal reasoning generally, that some assertion is affirmed to hold of every term; and unless the notion of every term is admitted, formal truths are impossible. (Russell 1903a, §44)

On this view, formality is essentially a matter of degree; “for all (x), Human(x) Ⱥ Mortal(x)” is a formal truth, but the propositional function can also be regarded as an instance of a further form, namely . The latter form has false as well as true instances. “For all (x), Human(x) Ⱥ Mortal(x)” is therefore not a formal truth, when judged by the criteria for formality which are associated with . For Russell, formality thus expresses no more than generality with

88 This is in effect the notion of a logically analytic proposition that was briefly discussed in section 5.3.3.2. 574 Ch. 5 Logic as the Universal Science II respect to a chosen set of constants, and logic stands out only because it possesses the highest degree of generality. It is this idea whose consequences will be explored below.

5.9.2 Russell on Valid Inference

“Symbolic Logic”, Russell writes in the Principles, §12, “is essentially concerned with reasoning in general, and is distinguished from various special branches of mathematics mainly by its generality”. The early Russell’s account of reasoning, though not developed in any rigorous manner, can be seen to accord with what one would expect from an advocate of the Bolzanian account of logic. In the Principles, the fullest treatment of inference is the following passage from §37:

The relation in virtue of which it is possible for us validly to infer is what I call material implication. [...] [I]n developing the consequences of our assumption as to implication, we were led to conclusions which do not by any means agree with what is commonly held concerning implication, for we found that any false proposition implies every proposition and any true proposition is implied by every proposition. [...] It would certainly not be commonly maintained that “2 + 2 = 4” can be deduced from “Socrates is a man,” or that both are implied by “Socrates is a triangle.” But the reluctance to admit such implications is chiefly due, I think, to preoccupation with formal implication, which is a much more familiar notion, and is really before the mind as a rule, even where material implication is explicitly mentioned. In inference from “Socrates is a man,” it is customary not to consider the philosopher who vexed the Athenians, but to regard Socrates merely as a symbol, capable of being replaced by any other man; and only vulgar prejudice in favour of true propositions stands in the way of replacing Socrates by a number, a table, or a plum-pudding. Nevertheless, wherever, as in Euclid, one particular proposition is deduced from another, material implication is involved, though as a rule the material implication may be regarded as a particular instance of some formal implication, obtained by giving some constant value to the variable or variables involved in the said formal implication. Ch. 5 Logic as the Universal Science II 575

Russell begins his account by claiming that material implication – that is, simple truth-preservation – is the relation which grounds valid inference. To this simple account, however, he soon adds the qualification that when one proposition is deduced from another, the two are, “as a rule”, connected by a formal implication. In the above quotation he does not say that the simple account must be modified; he merely states that a formal implication is involved “as a rule”. Elsewhere in the Principles, however, he seems to require that the further condition must be always fulfilled. Thus, in §45 he states that formal implication is involved in all rules of inference. If, now, inference is always backed up by a suitable logical law or rule, it follows that in an inference a material implication must always be an instance of a formal implication; and as he states in §17 that no demonstration is possible without principles of inference, i.e., certain formal implications, we may conclude that he accepted the modified view in the Principles. It is clear enough that some modification must be made to the original, simple account. That there is more to valid inference than mere ordinary truth-preservation is easily seen by considering some uncontroversial characteristics of reasoning. The very idea of reasoning, we may say, is to arrive at a judgment about the truth-value of a proposition on the basis of judgments about the truth-values of some other propositions. Given a proposition, we may know, or have good reasons to believe, that it is true, and seeing that something else follows from it, we have thereby found a good reason to think that the other proposition is true; or, we may simply pick a proposition, and try to determine what follows from it, what would be the case if it was true. The relation of material implication cannot account for these platitudes about reasoning. Whether material implication holds between two propositions, A and B, is something that depends solely upon their actual truth-values: A implies B only if either A is false or B is true. Knowing this much, however, does nothing to help us determine the truth-value of a proposition. If A is false, we know that it 576 Ch. 5 Logic as the Universal Science II materially implies every proposition; knowing this, though, does not enable us to determine the actual truth-value of any other proposition than A. On the other hand, if we know that A is true, then, given some other proposition, B, we can conclude “either B is true or else B is not implied by A”. Again, however, this is not a useful piece of information. Finally, if we know that B is true, we can conclude that it is implied by A, or any other proposition, but then we no longer need this additional piece of information to help us reach a conclusion about the truth-value of B; after all, we were supposed to infer the truth-value of a proposition on account of seeing that it stands at the receiving end of a relation of implication, and not the other way round. It is a further question whether Russell’s additional condition – that ordinary truth-preservation is something that can be regarded as holding generally – is the right one. There are very good reasons to think that it is not. It is of course what one would expect from an advocate of the Bolzanian account of logic, which operates with a simple quantificational account of generality.89 This yields the following characterization of validity or of “formal validity”:

(VAL) validity (or formal validity) equals truth-preservation + generality of an appropriate kind.

(VAL) means that an argument like

(*) the mission fails; therefore the mission fails or the mission succeeds is valid simply because every argument possessing the form P, therefore P Q is truth-preserving. As validity is in this way relativised to one or more constants which determine what counts as form, we can, with equal justification, say that, for example

(**) Kamal Haasan is human; therefore Kamal Haasan is mortal

89 See the end of section 5.5. Ch. 5 Logic as the Universal Science II 577 is valid with respect to the form x is human; therefore x is mortal. This is so because every argument possessing this form is truth-preserving (we may suppose). On a pure Bolzanian account of validity, (*) and (**) are structurally identical arguments. The only difference is that the latter is less general, which means that its underlying form has a narrower field of application than the one that belongs to (*). This explanation, however, seems wrong, and it seems wrong for precisely the same reason as the initial account. The point can be put as follows. The relation which grounds valid inference must be one that can be effectively recognised; if it is not, there can be no such thing as proving anything or deducing consequences from a given set of assumptions. Evidently, (**) does not qualify as a genuine consequence. That the conclusion is correct (supposing it is), is something that can be decided only by surveying the relevant instance, not by inferring it from the premise.90 An account of genuine consequence – valid inference – should be capable of distinguishing between (*) and (**). The pure Bolzanian account fails this test.91 This conclusion is not explicitly stated in the Principles. In a slightly later account of inference, found in “Necessity and Possibility”, it comes out quite clearly. In that paper Russell argues first that the possibility of inferring one proposition from another is based on the relation of material implication. He then points out that this does not quite suffice:

90 The point is not that Kamal Haasan is human; therefore Kamal Haasan is mortal does not qualify as a genuine piece of reasoning. No doubt, if we think that Mr. Haasan is mortal, we think so, because we have inferred it from the fact that he is human together with what we think we know about human beings; after all, we have good reasons to believe that all mortals are in fact mortal. The point is that this piece of reasoning is different from the mission fails; therefore the mission fails or it succeeds in a way that is connected with effective recognisability. 91 The points I have been making here are not new. They are familiar from Etchemendy’s (1990) criticisms of Tarski’s account of logical consequence. 578 Ch. 5 Logic as the Universal Science II

But in the practice of inference, it is plain that something more than implication must be concerned. The reason that proofs are used at all is that we can sometimes perceive that q follows from p, when we should not otherwise know that q is true; while in other cases “p implies q” is only to be inferred either from the falsehood of p of from the truth of q. In these other cases, the proposition “p implies q” serves no practical purpose; it is only when this proposition is used as a means of discovering the truth of q that it is useful. (1905a, p. 515)

The reason why more than material implication is needed is essentially the one given above: proofs are possible, because one can recognise that p implies q without knowing the actual truth-values of the propositions. Thus, something more than mere material implication is involved in what Russell calls “valid deduction” (id. p. 517). When, however, he turns to explaining what the further condition is, he does not invoke the notion of formal implication, which is the notion that he had used in the Principles. Instead, he refers to an epistemic condition, namely (self-)evidence. The passage quoted above continues:

Given a true proposition p, there will be some propositions q such that the truth of “p implies q” is evident, and thence the truth of q is inferred; while in the case of other true propositions, their truth must be independently known before we can know that p implies them. What we require is a logical distinction between these two cases. (id., p. 515; first italics added)

This further condition tallies well with the notion of effective recognisability introduced above. Russell, however, is not satisfied with the epistemic notion, as he thinks that what is really needed is a logical distinction between mere material implication and the relation underpinning valid deduction. On this score, however, he has very little to say by way of a genuinely illuminating account. What he does is in fact to invoke a primitive notion of law of deduction. He says that deduci- bility is to be understood through derivability: q is deducible from p only if it can be shown by means of explicitly formulated such laws that p implies q. Ch. 5 Logic as the Universal Science II 579

Russell thus gives a syntactic twist to his notion of logical validity, for now that notion is identified with derivability in some explicitly formulated deductive system (like the one given in Theory of Implica- tion). It is not difficult to say why he should end up saying such things about validity. His strategy, we have seen, presupposes a primitive notion of logical law. As his examples show – he mentions “if not-p is false, then p is true”, “if p implies not-q, then q implies not-p” and “if p implies q and q implies r, then p implies r” – these laws are just the formal implications of the Principles. As we have seen, an advocate of the Bolzanian account of logic has available to himself no genuinely explanatory criterion of logicality wit the help of which he could distinguish genuine consequences from spurious. When he re-states his characterization of the notion of law of deduction, he mentions “forms”: “The laws of deduction tell us that two propositions having certain relations of form (e.g. that one is the negation of the other) are such that one of them implies the other” (1905a, p. 515). “Form”, however, is not an explanatory notion in the context of the Bolzanian account of logic. As for the intuitive criterion of self-evidence, a natural suggestion would be to unpack it with the help of the notions of primitiveness and apriority. That is, when q is a logical consequence of p, the two are connected via a series of minimal inferential steps, each of which is (i) licensed by an evident law of deduction, and (ii) can be recognized as correct without consulting experience. The problem with this suggestion is that it is doubtful whether there is room for the latter notion in the early Russell’s metaphysics of propositions; if not, only the logical notion remains, but this is not explanatory, either.

5.9.3 Russell on Lewis Carroll’s Puzzle

Above I argued that the distinction between genuine consequence and mere truth-preservation is not drawn explicitly in the Principles. It seems, though, that Russell in fact came very close to stating it, even 580 Ch. 5 Logic as the Universal Science II if the conclusion is not expressed in so many words. The basic problem with the notion of formal implication can be put simply by asking: How do formal implications figure in an inference? They must be granted some role, for the validity of deduction is supposed to depend upon them: “The fundamental importance of formal implication is brought out by the consideration that it is involved in all the rules of inference” (1903a, §45). The problem with formal implications is brought to the fore by the puzzle about inference that Lewis Carroll made famous in his “What the Tortoise Said to Achilles” (Carroll 1895). As Carroll formulates it, the puzzle is about what a participant in a reasoning game may and may not (or must and need not) “grant”, and hence it is about the movements of the reasoning mind. In order to reach a conclusion, B, from the premises A and if A, then B, the reasoner must first grant the premises and then grant the conclusion. For this series of grantings to constitute an inference, it is necessary that there should be a justificatory relation between the different grantings. That is, it should be the case that the granting or acceptance of the conclusion is, logically speaking, conditional upon a granting of the premises; seeing that the justification is there is presumably what sets the reasoner’s mind in motion. Thus, the argument continues, the acceptance of the conclusion must wait until the reasoner has granted not only the two premises, but also that the conclusion follows from them logically. If, however, the result of this recognition is just another proposition – if A and if A, then B, then B – it should be included among the premises, and then the question would immediately arise as to how the reasoner gets from this new set of premises to the original conclusion, B. Repeating the above reasoning, it seems we must conclude that the reasoner is always in need of yet another proposition in order to bridge the gulf between the ever increasing set of premises and the original conclusion. Commenting on Carroll’s puzzle in the Princuples, §38, Russell argues that it shows that a distinction must be drawn between implies and therefore. The latter relation obtains between asserted propositions Ch. 5 Logic as the Universal Science II 581 and is associated with Russell’s version of modus ponens, the informal rule (4) of his propositional calculus: “A true hypothesis in an implication may be dropped, and the consequent asserted”. Once the reasoner has granted (asserted) A, and has also granted (asserted) that A implies B, he is in a position to grant (assert) B. Russell’s rule licenses this transition – apparently anyway – and is thus not just another statement to the effect that the relation of implication holds between certain kinds of entities. In Russell’s view, then, the puzzle is solved, once the distinction between logical rules and laws is recognized. All this is very reasonable, and it is certainly on the right track as a reply to Carroll’s puzzle. Nevertheless, Russell continues to insist that the validity of inference is grounded in suitable formal implications. How, then, is this grounding to be explained? Unfortunately for Russell, all he can say by way of an explanation is that the formal implication implies its instance, for instance:

(p)(q)(p (p q)) (mission succeeds (mission succeeds mission fails))

But this implication in no way contributes to the validity of the inference mission succeeds; therefore, mission succeeds or it fails. For the formal implications implies not only its instances but any true implication, for instance

(p)(q)(p (p q)) (Aristotle is mortal Socrates is mortal)

Clearly, a fact about disjunction-introduction in no way justifies us in inferring “Socrates is mortal” from “Aristotle is mortal”. The only alternative would be, in fact, to resort to a new formal implication, one to the effect that the original relation between a formal implication and its instance is just a special case of what holds generally. But as this is just another formal implication, it fails to ground the relation between “(p)(q)(p (p q))” and “mission succeeds (mission succeeds mission fails)”. In other words, one could derive the in- 582 Ch. 5 Logic as the Universal Science II stance from the general proposition, but the derivation could scarcely be used to show that the fact that the general principle is valid grounds the validity of inference. When it comes to accounting for the validity of inference, Rus- sell’s situation in the Principles is this. Validity is grounded in “principles of deduction”. These are simply (certain kinds of) formal implications. One’s first attempt at explaining how a formal implication figures in an inference is undoubtedly to say that it is one of the premises. Carroll’s puzzle, however, shows that this cannot be the case. The difficulty pinpointed by Carroll’s puzzle is to be sidestepped by postulating logical rules as distinct from logical laws. If, now, one continues to say – as Russell does – that validity nevertheless is a matter of there being suitable formal implications, how else is this to be understood except by assuming that a formal implication somehow grounds the rule of inference? For Russell this “somehow” involves either material implication – the formal implication implies its instance – or another formal implication – the relation of the formal implication to its instance is a special case of what holds generally. Neither of these options is satisfactory. Therefore, formal implications are not relevant to inference. And it seems that he comes close to making this point at the end of §45 of the Principles, where he argues that the fact that a formal implication does imply its instance, “if introduced at all, must simply be perceived”, and that it is often just as easy, and consequently just as legitimate, to perceive that the implication holds in a particular instance. The root problem in all this is of course the view that the “relation in virtue of which it is possible for us validly to infer is what I call material implication”. This is the official view in the Principles, and Russell maintains it (omitting what is stated at the end of §45), although he has at his disposal all the ingredients for an argument which decisively undermines the view. As we saw above, this conclusion is in effect accepted in “Necessity and Possibility”, and there it leads to the conclusion that in the case of valid inference the implica- Ch. 5 Logic as the Universal Science II 583 tion must be evident, requirement that comes close to what is found in §45 of the Principles. Russell is thus led to accepting of a primitive notion of logicality by considerations concerning both the content of logical propositions and the nature of valid inference. In the latter case, the only way genuinely valid inference can be distinguished from such as are spurious is to postulate a primitive notion of “law of deduction”.

5.9.4 The Bolzanian Account of Logic and Russell’s Ontology

It is easy to see that Russell understanding of logical validity complies with what is involved in the Bolzanian account of logic. In general, an advocate of the Bolzanian account resorts to some variant of the following explanation. For example, the reason why modus ponens is a valid form of inference is this:

if P, then Q; P/ Q is a valid form of inference, because no matter what propositions one puts in place of P and Q, the result is always a truth-preserving argument.

Clearly, the “always” which figures in this formulation expresses the sort of generality that goes together with logical constants. For given nonstandard choices of constants, a structurally identical explanation may be given for the validity of an inference schema that our intuition declares (logically) invalid. “Logicality” and “logical constant” thus turn out to be primitive notions. But this is just the conclusion that we reached in our discussion of Russell’s views on valid inference. These views and Russell’s conclusions are exactly what one would expect from an advocate of the Bolzanian account: the natural explication of logical constanthood is one that makes use of a purely descriptive (or quantificational) notion of generality. But as there are different degrees of generality, this notion does not offer an independent criterion for logicality. 584 Ch. 5 Logic as the Universal Science II

In Russell’s case however, we can say more than just that he in fact accepted the Bolzanian account of logic. For there are good reasons to think that it was the only account available to him. The point can be argued for by considering the intuitive notion of logical consequence from Russell’s point of view. It is a familiar point from literature on logicality and logical constants that notions of logical (or genuine) consequence can be divided into three main types: modal, formal and epistemic;92 here Ƅ is a set of premises and Ɩ is the conclusion:

Modal accounts: Ɩ is a consequence of Ƅ if and only if it is impossible for every member of Ƅ to be true and Ɩ to be false;

Formal accounts: Ɩ is a consequence of Ƅ if and only if there is no argument with the same logical form with true premises and a false conclusion;

Epistemic accounts: Ɩ is a consequence of Ƅ if and only if the link between Ɩ and Ƅ is apriori recognizable (which, when unpacked, might yield “if and only if it would be irrational to accept Ƅ and reject Ɩ”).

Let us consider each of these in turn. The first one is clearly out of question for Russell. There is simply no way within the early Russell’s metaphysics to explain the relevant notion of possibility in an acceptable way. In the Principles, the point is put as follows: “There seems to be no true proposition of which there is any sense in saying that it might have been false. One might as well say that redness might have been a taste and not a colour. What is true, is true; what is false is

92 See, for example, Hanson (1997) and Shapiro (1998). Ch. 5 Logic as the Universal Science II 585 false, and concerning fundamentals, there is nothing more to be said” (§430). Indeed, in the same section we find him asserting the following view about logic: “Everything is in a sense a mere fact. A proposition is said to be proved when it is deduced from premises; but the premises, ultimately, and the rule of inference, have to be simply assumed. Thus any ultimate premise is, in a certain sense, a mere fact.” As we saw in discussing Russell’s version of the Bolzanian account of logic, he admitted that there are legitimate senses of necessary, possible, and impossible. Their primary application, however, is to propositional functions and not to propositions; a propositional function is necessary, if it yields only true propositions, possible, if at least one of its values is a true proposition, and impossible, if all the propositions which are its values are false. It is evident that this account of modalities – which, incidentally, was a stable part of his logical atom- ism93 – does not accord at all well with our modal common sense; on this view, for example, and are both necessary in exactly the same sense. Russell does recognize other possible uses for modal notions, but none of them can be used to delineate an informative sense in which logical consequence (and logical truth) could be described as necessary. For instance, in “Necessity and Possibility” Russell states that we may regard as necessary propositions that are analytic, where analyticity is characterized in terms of derivability from logical laws (1905a, pp. 516-7). Here the explanans contains reference to a primitive notion of logical law, which makes the notion of necessity thus delineated useless in the present context. The epistemic account is potentially more promising. As we noted above, Russell himself uses it on one occasion to distinguish deduci- bility from implication. But as I indicated, it is questionable whether there is room for a notion of apriority in Russell’s metaphysics of propositions. The short argument for this view is as follows:

93 The quantificational account of modality is endorsed, for example, in Russell (1912-13, pp. 175-6), (1914b, p. 109) and (1918b, p. 231). For a detailed discussion, see Dejnozka (1999). 586 Ch. 5 Logic as the Universal Science II

No matter what proposition p is and no matter what constituents p has, every cognitive relation that the epistemic subject may have to p is based on the simple relation of grasping. Hence, even if p is a proposition of logic and its constituents are logical constants, to cognize p is still to grasp its content. Therefore, there is no difference between the cognition that is apriori and cognition that is aposteriori.

Hylton characterizes this view of knowledge as follows: “[knowledge] is a direct and immediate relation between a mind which is passively receptive and an object which is unaffected by being known” (1990a, p. 112). As he points out (ibid.), this model of direct acquaintance is intended to apply to all kinds of entities. In order to explain apriority, Russell could resort to some objective, rather than cognitive, construal of the notion. That is, he could maintain that apriori propositions differ from other propositions on account of what kinds of constituents they may have. However, this distinction itself is problematic in the context of Russell’s metaphysics. My left foot is a Russellian term (or so we may suppose for the sake of argument); the reason why propositions about it are aposteriori is that my left foot stands in a certain relation to the term existence. On the other hand, the reason why the propositions in which some “abstract” object occurs as the logical subject are typically apriori is that existence is not a constituent of those propositions. To put it another way, my left foot and an object that is abstract are both entities that have being (no doubt, this sounds somewhat strained, but we may ignore this problem in the present context); the difference between them is that one of them bears a certain relation to existence, while the other does not, and existence itself is just one more term among others. Considered in this way, the propositions of logic are probably found on the abstract side, but it is not clear whether the distinction between apriori and aposteriori, if drawn in this way, marks any genuine difference between the items to which it applies. Ch. 5 Logic as the Universal Science II 587

At least, it is difficult to see how it could provide a useful and genuinely illuminating criterion for logical validity. The only remaining possibility of logical validity, i.e., the only account that is consistent with Russell’s fundamental philosophical commitments, is the family of formal accounts. The fundamental notions in Russell’s metaphysics are those of proposition and constituent of proposition. Since the latter notion is perfectly general – the notion of term does not discriminate but includes everything – the non- distinction between content and form follows immediately. Insofar as there is modality, it is to be expressed in purely quantificational terms. Thus, logical truth and validity, notions that prima facie have a modal dimension to them, are to be captured by resorting to (unrestricted) generality. As we have seen, though, this move relies either explicitly or implicitly on a primitive notion of logicality or logical constant.

5.10 The Apriority of Logic

At the most general level, the differences between Kant and Russell are differences between philosophers advocating two conceptions of objectivity. Kant’s conception can be described as intersubjective – hence the emphasis on experience and its conditions – while Russell’s is ontological – hence the notion of proposition. It could be said that the difference between the two philosophers does not lie in the fact that they offer competing explanations for a range of phenomena. Rather the disagreement is about the proper explanandum. The point can be formulated in terms of the concept of truth. Kant wants to explain how there can be such thing as judgments-taken-to-be-true, whereas Russell wants to understand truth itself.94

94 The contrast, I believe, is also to be found between Kant and Frege, though the matter is a good deal less clear-cut in this case. Yet another formulation of the contrast between Kant and Russell would be to say – and here I refer to the discussion in section 3.6.3.3.2 – that for Kant the conditions of objectivity are conditions for there being thoughts of a certain particular 588 Ch. 5 Logic as the Universal Science II

Thus, when it comes to explaining how propositions can possess a range of properties (crucially, those that they possess in virtue of being apriori), Kant sees this as equivalent to explaining how judgments, that is, our judgings-as-true, can possess certain properties. For Rus- sell, by contrast, this is never a question, as it signals merely a deep confusion on the part of the philosopher who asks it. Seen in this way, the differences between the two philosophers are about as deep as they can be. For the dispute is not over what constitutes correct explanans but about what constitutes correct explanandum. Russell’s criticisms of the relativized model of the apriori give expression to this difference of perspectives. At the end of chapter 3 the claim was endorsed that there is a clear sense in which Russell fails to address the critical question, How is pure logic possible? As long as our perspective on this question is Kantian, the claim is quite correct: the early Russell showed little or no interest in questions that are epistemic. And for the Kantian, the question is epistemic. To ask, how pure logic is possible, is to ask, in effect, two questions. Firstly, it is to ask, what role does pure logic have in our cognitive life? Secondly, it is to ask, what must be the case for it to fulfil that role? Given this perspective, the notion of apriority assumes a special role, to wit, to help identify the concepts and propositions that first make “objective thought” possible. It is also this perspective that gives the concepts and propositions which have been identified in the course of transcendental reflection whatever justification they may have; very briefly, their justification lies in the very special role that they play in our cognitive life. For Russell, by contrast, the fundamental question is essentially simpler. To ask kind (like mathematical judgments or judgments about spatio-temporal entities which stand in the relation of causal interaction). These thoughts are essentially thoughts-for-us, although it is only at the level of transcendental reflection that this dependence becomes evident. Russell, by contrast, recognizes no such hidden parameter; for him the objects of thought are what they are independently of the cognitive relation in which we may stand to them. Ch. 5 Logic as the Universal Science II 589 for an explanation of how pure logic is possible is to ask a perfectly straightforward question, namely: what is the content of the propositions of pure logic? Or, what are the constituents of the propositions of logic? To answer this question is to articulate the ontology of logic. Logic has ontological implications, and these can be found out by reflecting on the content of the propositions of logic. This is the ultimate reference-point for the sort investigation that Russell sees as the proper task for philosophy. To be sure, this in no way excludes subsequent investigation into the epistemology of logic.95 As Russell sees it, however, the Kant strategy, with its emphasis on cognitive processes, fails to address the more fundamental questions concerning concepts and content. Better yet, the charge is that Kant has confused questions about X with questions about our cognition of X, questions that are plainly different.96 97 [1] Reflection on the content of the concept of truth shows that the properties of truth can by captured only by taking it to be an ontological notion, a property of propositions. This applies to the propositions of logic as well. They are true, not because we cannot but take them to be true (this was the Kantian explanatory strategy), but simply in virtue of the content of the concept of truth.

95 Although it must be said that Russell’s perspective positively encourages the misconception that epistemic questions – questions about the relation of the knowing subject to the objects of knowledge – are always psychological questions. 96 From Russell’s standpoint, moreover, there can be little doubt as to who is right here. To use an example that was discussed at length in section 3.6.3.2, it is plain that propositions about space and propositions about our thoughts about space have different constituents, something that can be seen by reflecting on the content of the relevant propositions. 97 This means also that Russell has no use for the notion of constitutive rule, which plays an important role in the Kantian epistemology of the apriori; a concept or proposition that is constitutively apriori is one that is constitutive of a practice. The notion of practice, however, is one in which the early Russell has no interest. 590 Ch. 5 Logic as the Universal Science II

[2] Universality. The proposition of logic are universal, that is to say, maximally general, not because they dictate the limits of experience or of thought, but simply in virtue of the content of the concept absolutely everything, i.e., in virtue of what it is for something to be about everything. [3] Necessity. As we have seen, there is no room in Russell’s metaphysics for a substantive notion of necessity. Russell is thus committed to the view – possibly offending our intuitions about logic and logical validity – that the propositions of logic are characterized by nothing more than plain truth. An advocate of the Bolzanian account of logic, he is committed to the view that “there is no such comparative and superlative of truth as is implied by the notion of contingency and necessity” (1905a, p. 520). Now, one might argue that this neglect (if such it be) of the modal dimension constitutes evidence for a Kantian conception of apriority and against the Russellian one. Of course, from Russell’s point of view this is just an error; the support that Kant’s theory seems to gain from considerations of necessity is entirely spurious; for Kant’s explanation of necessity is “radically vicious” (1903a, §431). Kant can explain necessity only by grounding it in something factual, namely our cognition. This explanatory move, however, is mistaken on two counts. Firstly, to derive the necessity of something from what is merely factual is to undermine that something’s claim to necessity. Secondly, and more importantly, to explain necessity by reducing it to what is necessary-for- us is to misconstrue the import of the concept of necessity. By hindsight, Russell’s model of the synthetic apriori appears to suffer from multiple deficiencies. Apart from the criticisms that may be levelled against the idea that the propositions of logic are synthetic propositions with genuine content, there is the charge that the model does not really address the many questions raised by the category of the apriori at all. Whether we see this charge as justified depends at least in part on whether our sympathies are with Kant or with Russell, when it comes to judging the controversy between their respective explanatory strategies. However, subsequent developments in Rus- Ch. 5 Logic as the Universal Science II 591 sell’s philosophy (and elsewhere) suggest a broader perspective. Firstly, Russell’s slightly later remarks on the “regressive method” (Russell 1907b) may be taken to suggest that the mathematical knowledge should be seen as a species of aposteriori, rather than apriori knowledge. Secondly, the distinction between pure and applied mathematics (with logic associated with the former) may suggest that the traditional picture of what is involved in and implied by the purity of mathematical knowledge may stand in need of radical revision.

5.11. Concluding Remarks

In this work I have examined Russell’s early conception of logic from two quite different interpretative perspectives. According to the van Heijenoort interpretation, Russell and other advocates of the universalist conception of logic think of their discipline as a universal (and unique) framework in which all correct thought – or at any rate all deductive reasoning – takes place. This view, it is argued, has far- reaching consequences for a number of fundamental logical issues. Briefly, the main interpretative claim is that a universalist logician is bound adopt a sceptical attitude towards a semantic (model-theoretic) approach to logic. When it comes to the van Heijenoort interpretation, I have argued for two conclusions. Firstly, its advocates tend to ignore the distinction between logic as a science – roughly, logic as the principles of correct reasoning – and a calculus for logic – a particular codification of such principles. Once this distinction is observed and put to use, most of the consequences that the proponents of the van Heijenoort interpretation have derived from the initial contrast between universalist and model-theoretic perspectives are seen to fail. Secondly, I have argued that, insofar as these consequences are correctly attributable to Russell, they should be traced to his metaphysics of propositions; for Russell, logic is not primarily a matter of language or linguistic structure, but belongs firmly to the sphere of being. Even here, how- 592 Ch. 5 Logic as the Universal Science II ever, reflection on his actual logical practice suggests that one had better be cautious in forming conclusions about Russell’s commitment in the philosophy of logic. On the positive side, I have argued that we can gain deeper understanding of Russell’s philosophy of logic by examining it in its proper historical and philosophical context. This context, which is explored in detail, is constituted by Russell’s anti-Kantianism and his view that logic is a synthetic and apriori science. Examination of the contrast between Kant’s and Russell’s views on the nature of formal logic leads to an articulation of what I call the Bolzanian account of logic. This perspective, I submit, is historically more correct than the picture to which the van Heijenoort interpretation gives rise. It also helps us to articulate a minimal sense in which the early Russell’s views are correctly describable as “universalist”. This minimal sense, however, is no more than a relatively simple and straightforward consequence of Russell’s metaphysics of propositions and the Bolzanian account of logic. Bibliography

Works by Bertrand Russell

(1895) “Review of Heymans, Die Gesetze und Elemente des wissenschaftlichen Denkens”, Mind, n.s., 4, 245-249. Reprinted in Russell (1983), 239-255. (1896a) “Review of Hannequin, Essai critique sur l’hypothese des atomes dans la science contemporaine”, Mind, n.s., 5, 410-417. Reprinted in Russell (1990), 35-43. (1896b) “On Some Difficulties of Continuous Quantity”, in Russell (1990), 44-58. (1896-98) “Various Notes on Mathematical Philosophy”, in Russell (1990), 6-28. (1897a) “Review of Couturat, De l’infini mathématique”, Mind, n.s., 6, 112-119. Reprinted in Russell (1990), 59-67. (1897b) “On the Relations of Number and Quantity”, Mind, n.s., 6, 326-341. Reprinted in Russell (1990), 68-82. (1897c) Essay on the Foundations of Geometry. With a new foreword by Morris Kline. New York: Dover. 1956. (1898a) “An Analysis of Mathematical Reasoning”, in Russell (1990), 155-238. (1898b) “Are Euclid’s Axioms Empirical?” Originally published in French as “Les Axiomes propres à Euclide, sont-ils empiriques?, Revue de métaphysique et de morale, 6, 759-776. English translation, from the French copy-text by G. H. Moore and N. Griffin, published in Russell (1990), 322-339. (1899a) “The Axioms of Geometry”. Translated in part and appeared in French as “Sur les axiomes de la géometrie”, Revue de métaphysique et de morale, 7, 684-707. English text compiled from Russell’s English manuscript and the French text, translated by G. H. Moore and N. Griffin, published in Russell (1990), 390-418. (1899b) “The Classification of Relations”, in Russell (1990), 136- 146. (1899c) “The Fundamental Ideas and Axioms of Mathematics”, in Russell (1990), 261-305. 594 Bibliography

(1899-1900) “The Principles of Mathematics, Draft of 1899-1900”, in Russell (1993), 3-180. (1900) A Critical Exposition of the Philosophy of Leibniz. Cambridge: Cambridge University Press. (1901a) “Recent Work on the Foundations of Mathematics”. First published in International Monthly, 4, 83-101. Reprinted under the title “Mathematics and the Metaphysicians” as chapter 4 in Russell (1918a); reprinted under the original title in Russell (1993), 363-379. References to the latter. (1901b) “Recent Italian Work on the Foundations of Mathematics”, in Russell (1993), 350-62. (1901c) “Is Position in Time and Space Absolute or Relative?”, Mind, n.s., 10, 293-317. Reprinted in Russell (1993), 259- 284. References to the latter. (1901d) “Part I of the Principles, Draft of 1901, in Russell (1993), 181-208. (1901e) “Sur la logique des relations avec des applications à la théorie des séries”, Revue de mathématiques, 7, 115-148. Published in English as “The Logic of Relations” in Russell (1956b), 1-38. Reprinted in Russell (1993), 310-349. (1903a) The Principles of Mathematics. With a new Introduction. London: George Allen and Unwin. 1937. (1903b) “Recent Work on the Philosophy of Leibniz”, Mind, n.s., 12, 177-201. Reprinted in Russell (1994), 537-561. (1903c) “On the Meaning and Denotation of Phrases”, in Russell (1994), 283-296. (1903d) “Dependent Variables and Denotation”, in Russell (1994), 297-304. (1903e) “Points about Denoting”, in Russell (1994), 305-313. (1903f) “On Meaning and Denotation”, in Russell (1994), 314-358. (1904a) “Meinong’s Theory of Complexes and Assumptions”, Mind, n.s. 13, 204-219; 336-354; 509-524. Reprinted in Russell (1973), 21-76; (1994), 431-474. (1904b) “Non-Euclidean Geometry”, Athenaeum, 124, 592-593. Reprinted in Russell (1994), 482-485. (1905a) “Necessity and Possibility”, in Russell (1994), 507-520. Bibliography 595

(1905b) “Review of Poincaré, Science and Hypothesis”, Mind, n.s., 14, 412-418. Reprinted as chapter III in Russell (1910); Russell (1994), 589-594. (1905c) “The Nature of Truth”, in Russell (1994), 490-507. (1905d) “On Fundamentals” in Russell (1994), 359-413. (1905e) “On Denoting”, Mind, n.s., 14, 479-493. Reprinted in Russell (1956b), 39-56; Russell (1994), 413-427. (1906a) “On ‘Insolubilia’ and Their Solution by Symbolic Logic”. First published in French as “Les Paradoxes de la Logique”, Revue de Métaphysique et de Morale, 14, 627-650. English version from the manuscript, published in Russell (1973), 190-214. (1906b) “On Some Difficulties in the Theory of Transfinite Numbers and Order Types”, Proceedings of the London Mathematical Society, ser. 2, vol. 4 (7 March 1906), 29-53. Reprinted in Russell (1973), 135-164. (1906c) “On the Substitutional Theory of Classes and Relations”, in Russell (1973), 165-189. (1906d) “The Theory of Implication”, American Journal of Mathematics, 28, 159-202. (1907a) “The Study of Mathematics”. New Quarterly, Nov. 1907, 29- 44. Reprinted in Russell (1910), Ch. III and Russell (1918a), Ch. IV. (1907b) “Regressive Method of Discovering the Premises of Mathematics”, in Russell (1973, 272-283). (1908) “Mathematical Logic as Based on the Theory of Types”, American Journal of Mathematics, 30, 222-262. Reprinted in Russell (1956b), 57-102. (1910) Philosophical Essays. London: Longmans, Green. (1911a) “The Philosophical Importance of Mathematical Logic”. First published in French as “L’Importance philosophique de la logistique”, Revue de métaphysique et de morale, 19, 281- 291. English translation by P. E. B. Jourdain (revised by B. Russell) first published in The Monist, 23 (1913), 481-493. Reprinted as “The Philosophical Implications of 596 Bibliography

Mathematical Logic” in Russell (1973), 284-294; reprinted under the original title in Russell (1992a), 32-40. (1911b) “On the Axioms of the Infinite and of the Transfinite”. First published in French as “Sur le axioms de l’infini et du transfini”, Société mathématique de France, Comptes rendues des séances, Paris, no 2. 22-35. English translation by I. Grattan- Guinness in Grattan-Guinness (1977), 161-174. English translation by A. Vellino in Russell (1992a), 41-53. References to the latter. (1912a) The Problems of Philosophy. London: Williams and Norgate. (1912b) “What is Logic?”, in Russell (1992a), 54-56. (1912-13) “On the Notion of Cause”. First Published in Proceedings of the Aristotelian Society, 13, 1-26. Reprinted as chapter 9 of Russell (1918a); reprinted in Russell (1992a), 190-210. (1914a) Our Knowledge of the External World. Chicago: Open Court. (1914b) “On Scientific Method in Philosophy”, in Russell (1918a), 96-120. (1918a) Mysticism and Logic and Other Essays. London: Longmans, Green & Co. Reissued under its original title in 1986 by Unwin Paperbacks. Reprinted with a new introduction by J. G. Slater. London: Routledge. 1994. (1918b) “The Philosophy of Logical Atomism”. First published in The Monist, 28, 495-527; 29, 32-63, 190-222, 345-380. Reprinted in Russell (1956b), 177-281; reprinted in Russell (1986), 157-244. (1919) Introduction to Mathematical Philosophy. London: George Allen and Unwin. (1924) “Logical Atomism”, in Contemporary British Philosophy: Personal Statements. First Series. Ed. by J. H. Muirhead. London: George Allen and Unwin, 357-383. Reprinted in Russell (1956b), 321-343. (1927) An Outline of Philosophy. London: George Allen and Unwin. (1929) “How I came by my Creed”, The Realist, London, I, no 6 (Sept. 1929), 14-21. First published as “What I believe”, in The Forum, September (82), 1929, 129-134. Bibliography 597

(1944) “My Mental Development”, in P. A. Schilpp (ed.) The Philosophy of Bertrand Russell. Library of Living Philosophers. Evanston, Illinois: Northwestern University Press, 1-20. (1945) A History of Western Philosophy, New York: Simon et Schuster. London: George Allen and Unwin. Unwin Paperbacks, 1984. (1950-52) “Is Mathematics Purely Linguistic?”, in Russell (1973), 295- 306. (1956a) Portraits from Memory, and Other Essays. London: George Allen and Unwin. (1956b) Logic and Knowledge. Ed. by R.C. Marsh. London: George Allen and Unwin. (1959) My Philosophical Development. London: George Allen and Unwin. (1967) The Autobiography of Bertrand Russell, Vol. I, 1872-1914. London: George Allen and Unwin. (1969) The Autobiography of Bertrand Russell, Vol. III, 1944-1967. London: George Allen and Unwin. (1973) Essays in Analysis. Ed. by D. Lackey, New York: George Braziller. (1983) The Collected Papers of Bertrand Russell, Vol. 1, Cambridge Essays 1888-99. Edited by K. Blackwell, A. Brink, N. Griffin, R. A. Rempel, J. G. Slater. London: George Allen and Unwin. (1984) The Collected Papers of Bertrand Russell, Vol. 7, The Theory of Knowledge: the 1913 Manuscript. Ed. by E. R. Eames in collaboration with K. Blackwell. London: George Allen and Unwin. (1986) The Collected Papers of Bertrand Russell, Vol. 8, The Philosophy of Logical Atomism and Other Essays: 1914-19. Ed. by J. G. Slater. London: George Allen and Unwin. (1990) The Collected Papers of Bertrand Russell, Vol. 2, Philosophical Papers 1896-99. Edited by N. Griffin and A. C. Lewis. London: Unwin, Hyman. (1992a) The Collected Papers of Bertrand Russell, Vol. 6, Logical and Philosophical Papers 1909-13. Edited by J.G. Slater. London, Boston and Sydney: Allen and Unwin. 598 Bibliography

(1992b) The Selected Letters of Bertrand Russell, Vol. I, The Private Years. Edited by N. Griffin. London: Penguin. (1993) The Collected Papers of Bertrand Russell, Vol. 3, Towards the Principles of Mathematics. Edited by G. H. Moore, London: Routledge. (1994) The Collected Papers of Bertrand Russell, Vol. 4, The Foundations of Mathematics. Edited by A.Urquhart. London and New York: Routledge.

Works by Other Authors

Adams, R. (1995) Leibnz: Determinist, Theist, Idealist. Oxford: Oxford University Press. Allison, H. (1973) Kant-Eberhard Controversy. English translation together with supplementary materials and a historical-analytical introduction of Immanuel Kant’s “On a discovery according to which any new critique of pure reason has been made superfluous by an earlier one. Baltimore: Johns Hopkins University Press. – (1983) Kant’s Transcendental Idealism, an Interpretation and Defence. New Haven and London: Yale University Press. Alnes, J. H. (1998) Frege on logic and logicism. Oslo: University of Oslo, Department of Philosophy. Andersson, S. (1994) In Quest of Certainty. Bertrand Russell’s Search for Certainty in religion and mathematics up to “Principles of Mathematics (1903). Studia philosophiae religionis 18. Stockholm: Almqvist & Wicksell. Ayer, A. J. (1936) Language, Truth and Logic. London: Gollancz. Badesa. C. (2004) The Birth of Model Theory. Löwenheim’s Theory in the Frame of the Theory of Relatives. Princeton: Princeton University Press. Baker, G. P. & P. M. S. Hacker (1989) “Frege’s Anti-Psychologism”, in M. Notturno (ed.) Perspectives on Psychologism. Leiden: E. J. Brill, 175-122. Baldwin, T. (1990) G. E. Moore. London, New York: Routledge. Bibliography 599

Beck, L. W. (1955) “Can Kant’s Synthetic Judgments be made Analytic?”, Kant-Studien, 45, 168-181. Reprinted in R. F. Chadwick and C. Cazeaux (eds.) Immanuel Kant. Critical Assessments, Vol. 2: Kant’s Critique of Pure Reason. London & New York: Routledge: 347-362. References to the latter. Bell, D. (1999) “The Revolution of Moore and Russell: A Very British Coup?”, in A. O’Hear (ed.) German Philosophy Since Kant. Royal Institute of Philosophy Supplement, 44. Cambridge. Cambridge University Press, Benacerraf, P. (1973) “Mathematical Truth”, The Journal of Philosophy, 70, 661-679. – (1981) “Frege, the Last Logicist”, in P. French et al. (eds.) Midwest Studies in Philosophy VI. Minneapolis, Minneapolis University Press, 17-35. Benacerraf, P. and H. Putnam (eds.) (1964) Philosophy of Mathematics. Cambridge: Cambridge University Press. Berg, J. (1962). Bolzano’s Logic. Stockholm Studies in Philosophy 2. Stockholm: Almqvist & Wiksell. Berkeley, G. Treatise Concerning the Principles of Human Knowledge. In The Works of George Berkeley, bishop of Cloyne. Edited by A. A. Luce and T. E. Jessop. Vol. 2: The Principles of Human Knowledge. Edited by T. E. Jessop. Edinburgh: Thomas Nelson and Sons Ltd. 1949. Bernays, P. (1967) “Hilbert, David”, in P. Edwards (ed.) The Encyclopedia of Philosophy, Vol. 3. New York: MacMillan, 496-504. Blackburn, Simon (1986) “Morals and Modals”, in G. MacDonald and C. Wright (eds.) Fact, Science and Morality: essays on A. J. Ayer’s ‘Language, Truth and Logic’. Oxford: Blackwell, 119-141. Boghossian, P. (2000) “Knowledge of Logic”, in Boghossian, P. & C. Peacocke (eds.) (2000), 229-254. Boghossian, P. & C. Peacocke (eds.) (2000), New Essays on the A Priori. Oxford: Oxford University Press. Bolzano, B. (1810) Beiträge zu einer begründeteren Darstellung der Mathematik. Prag. A Selection translated into English by S. Russ as Contributions to a better-grounded presentation of mathematics in Ewald (1996), 174-224. 600 Bibliography

– (1817) Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetztes Resultat gewähren, wenigstens eine reelle Wurzel der Gleichung liege, Prag. Translated into English by S. Russ as Purely Analytic Proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation, in Ewald (ed.) (1996), 225-248. – (1837) Wissenschaftslehre. Herausgegeben von Wolfgang Schultz. Leipzig: Felix Meiner 1929-31. A selection translated into English as B. Bolzano, Theory of Science. Edited, with an Introduction, by J. Berg. Translated from the German by B. Terrell. Dordrecht: D. Reidel Publishing Company. 1973. Bradley, F. H. (1883) The Principles of Logic, Vol. I. Second edition. Revised with commentary and terminal essays. Oxford: Oxford University Press. 1922. – (1893) Appearance and Reality. Oxford: Oxford University Press. Brittan, G. (1978) Kant’s Theory of Science. Princeton, N. J.: Princeton University Press. Broad. C. D. (1978) Kant: an Introduction. Ed. by C. Lewy. Cambridge: Cambridge University Press. Brook, A. (1992) “Kant’s A Priori Methods for Recognizing Necessary Truths”. In P. Hanson and Br. Hunter (eds.) Canadian Journal of Philosophy, Suppl. vol. 18, 215-52. Available online at http://http-server.carleton/~abrook/KT-APRIO.htm Buchdahl, G. (1969) Metaphysics and the Philosophy of Science. Cambridge, Mass.: MIT Press. Burge, T. (1997) “Frege on knowing the third realm”, in Tait, W. (ed.) (1997), 1-18. – (1998a) “Frege on Apriority”, in Boghossian, P. and C. Peacocke (eds.) (2001), 11-42. – (1998b) “Frege on Knowing the Foundation”, Mind, n.s., 107, 305- 347. Butts, R. E. (1981) “Rules, Examples and Constructions: Kant’s Philosophy of Mathematics”, Synthese, 47, 257-288. Byrd, M (1989) “Russell, Logicism and the Choice of Logical Constants”, Notre Dame Journal of Formal Logic, 30, 343-361. Bibliography 601

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